togethercomputer/RedPajama-Data-1T · Datasets at Hugging Face (2024)

\section{Introduction}Let $G$ be a simple undirected graph with the \textit{vertex set} $V(G)$ and the \textit{edge set} $E(G)$. A vertex with degree one is called a \textit{pendant vertex}. The distance between the vertices $u$ and $v$ in graph $G$ is denoted by $d_G(u,v)$. A cycle $C$ is called \textit{chordless} if $C$ has no \textit{cycle chord} (that is an edge not in the edge set of $C$ whose endpoints lie on the vertices of $C$).The \textit{Induced subgraph} on vertex set $S$ is denoted by $\langle S\rangle$. A path that starts in $v$ and ends in $u$ is denoted by $\stackrel\frown{v u}$.A \textit{traceable} graph is a graph that possesses a Hamiltonian path.In a graph $G$, we say that a cycle $C$ is \textit{formed by the path} $Q$ if $ | E(C) \setminus E(Q) | = 1 $. So every vertex of $C$ belongs to $V(Q)$.In 2011 the following conjecture was proposed:\begin{conjecture}(Hoffmann-Ostenhof \cite{hoffman})Let $G$ be a connected cubic graph. Then $G$ has a decomposition into a spanning tree, a matching and a family of cycles.\end{conjecture}Conjecture \theconjecture$\,$ also appears in Problem 516 \cite{cameron}. There are a few partial results known for Conjecture \theconjecture. Kostochka \cite{kostocha} noticed that the Petersen graph, the prisms over cycles, and many other graphs have a decomposition desired in Conjecture \theconjecture. Ozeki and Ye \cite{ozeki} proved that the conjecture holds for 3-connected cubic plane graphs. Furthermore, it was proved by Bachstein \cite{bachstein} that Conjecture \theconjecture$\,$ is true for every 3-connected cubic graph embedded in torus or Klein-bottle. Akbari, Jensen and Siggers \cite[Theorem 9]{akbari} showed that Conjecture \theconjecture$\,$ is true for Hamiltonian cubic graphs.In this paper, we show that Conjecture \theconjecture$\,$ holds for traceable cubic graphs.\section{Results}Before proving the main result, we need the following lemma.\begin{lemma}\label{lemma:1}Let $G$ be a cubic graph. Suppose that $V(G)$ can be partitioned into a tree $T$ and finitely many cycles such that there is no edge between any pair of cycles (not necessarily distinct cycles), and every pendant vertex of $T$ is adjacent to at least one vertex of a cycle. Then, Conjecture \theconjecture$\,$ holds for $G$.\end{lemma}\begin{proof}By assumption, every vertex of each cycle in the partition is adjacent to exactly one vertex of $T$. Call the set of all edges with one endpoint in a cycle and another endpoint in $T$ by $Q$.Clearly, the induced subgraph on $E(T) \cup Q$ is a spanning tree of $G$. We call it $T'$. Note that every edge between a pendant vertex of $T$ and the union of cycles in the partition is also contained in $T'$. Thus, every pendant vertex of $T'$ is contained in a cycle of the partition. Now, consider the graph $H = G \setminus E(T')$. For every $v \in V(T)$, $d_H(v) \leq 1$. So Conjecture \theconjecture$\,$ holds for $G$. \vspace{1em}\end{proof}\noindent\textbf{Remark 1.}\label{remark:1}Let $C$ be a cycle formed by the path $Q$. Then clearly there exists a chordless cycle formed by $Q$.Now, we are in a position to prove the main result.\begin{theorem}Conjecture \theconjecture$\,$ holds for traceable cubic graphs.\end{theorem}\begin{proof}Let $G$ be a traceable cubic graph and $P : v_1, \dots, v_n$ be a Hamiltonian path in $G$. By \cite[Theorem 9]{akbari}, Conjecture A holds for $v_1 v_n \in E(G)$. Thus we can assume that $v_1 v_n \notin E(G)$. Let $v_1 v_j, v_1 v_{j'}, v_i v_n, v_{i'} v_n \in E(G)\setminus E(P)$ and $j' < j < n$, $1 < i < i'$. Two cases can occur:\begin{enumerate}[leftmargin=0pt,label=]\item\textbf{Case 1.}Assume that $i < j$. Consider the following graph in Figure \ref{fig:overlapping} in which the thick edges denote the path $P$. Call the three paths between $v_j$ and $v_i$, from the left to the right, by $P_1$, $P_2$ and $P_3$, respectively (note that $P_1$ contains the edge $e'$ and $P_3$ contains the edge $e$).\begin{figure}[H] \begin{center} \includegraphics[width=40mm]{engImages/overlapping.pdf} \caption{Paths $P_1$, $P_2$ and $P_3$} \label{fig:overlapping} \end{center}\end{figure}If $P_2$ has order $2$, then $G$ is Hamiltonian and so by \cite[Theorem 9]{akbari} Conjecture \theconjecture$\,$ holds. Thus we can assume that $P_1$, $P_2$ and $P_3$ have order at least $3$. Now, consider the following subcases:\\\begin{enumerate}[leftmargin=0pt,label=]\label{case:1}\item \textbf{Subcase 1.} There is no edge between $V(P_r)$ and $V(P_s)$ for $1 \leq r < s \leq 3$. Since every vertex of $P_i$ has degree 3 for every $i$, by \hyperref[remark:1]{Remark 1}$\,$ there are two chordless cycles $C_1$ and $C_2$ formed by $P_1$ and $P_2$, respectively.Define a tree $T$ with the edge set$$ E\Big(\langle V(G) \setminus \big(V(C_1) \cup V(C_2)\big) \rangle\Big) \bigcap \big(\bigcup_{i=1}^3 E(P_i)\big).$$Now, apply \hyperref[lemma:1]{Lemma 1} $\,$for the partition $\{T, C_1, C_2\}$.\\\item \textbf{Subcase 2.}\label{case:edge}There exists at least one edge between some $P_r$ and $P_s$, $r<s$. With no loss of generality, assume that $r=1$ and $s=2$. Suppose that $ab \in E(G)$, where $a \in V(P_1)$, $b \in V(P_2)$ and $d_{P_1}(v_j, a) + d_{P_2}(v_j, b)$ is minimum.\begin{figure}[H] \begin{center} \includegraphics[width=40mm]{engImages/ab.pdf} \caption{The edge $ab$ between $P_1$ and $P_2$} \label{fig:ab} \end{center}\end{figure}Three cases occur: \\(a) There is no chordless cycle formed by either of the paths $\stackrel\frown{v_j a}$ or $\stackrel\frown{v_j b}$. Let $C$ be the chordless cycle $\stackrel\frown{v_j a}\stackrel\frown{ b v_j}$. Define $T$ with the edge set$$ E\Big(\langle V(G) \setminus V(C)\rangle\Big) \bigcap \big(\bigcup_{i=1}^3 E(P_i)\big).$$Now, apply \hyperref[lemma:1]{Lemma 1} $\,$for the partition $\{T,C\}$.\\(b) There are two chordless cycles, say $C_1$ and $C_2$, respectively formed by the paths $\stackrel\frown{v_j a}$ and $\stackrel\frown{v_j b}$. Now, consider the partition $C_1$, $C_2$ and the tree induced on the following edges,$$E\Big(\langle V(G) \setminus \big(V(C_1) \cup V(C_2)\big) \rangle\Big) \; \bigcap \; E\Big(\bigcup_{i=1}^3 P_i\Big),$$and apply \hyperref[lemma:1]{Lemma 1}.\\(c) With no loss of generality, there exists a chordless cycle formed by the path $\stackrel\frown{v_j a}$ and there is no chordless cycle formed by the path $\stackrel\frown{v_j b}$.First, suppose that for every chordless cycle $C_t$ on $\stackrel\frown{v_j a}$, at least one of the vertices of $C_t$ is adjacent to a vertex in $V(G) \setminus V(P_1)$.We call one of the edges with one end in $C_t$ and other endpoint in $V(G) \setminus V(P_1)$ by $e_t$. Let $v_j=w_0, w_1, \dots, w_l=a$ be all vertices of the path $\stackrel\frown{v_j a}$ in $P_1$. Choose the shortest path $w_0 w_{i_1} w_{i_2} \dots w_l$ such that $0 < i_1 < i_2 < \dots < l$.Define a tree $T$ whose edge set is the thin edges in Figure \ref{fig:deltaCycle}.\\Call the cycle $w_0 w_{i_1} \dots w_l \stackrel\frown{b w_0}$ by $C'$. Now, by removing $C'$, $q$ vertex disjoint paths $Q_1, \dots, Q_q$ which are contained in $\stackrel\frown{v_j a}$ remain. Note that there exists a path of order $2$ in $C'$ which by adding this path to $Q_i$ we find a cycle $C_{t_i}$, for some $i$. Hence there exists an edge $e_{t_i}$ connecting $Q_i$ to $V(G) \setminus V(P_1)$. Now, we define a tree $T$ whose the edge set is,$$\quad\quad\quad \bigg( E\Big(\langle V(G) \setminus V(C') \rangle \Big)\; \bigcap \; \Big(\bigcup_{i=1}^3 E(P_i)\Big) \bigg) \bigcup \Big(\big\{e_{t_i} \mid 1 \leq i \leq q \big\} \Big).$$Apply \hyperref[lemma:1]{Lemma 1} $\,$for the partition $\{T,C'\}$.\\\begin{figure}[H] \begin{center} \includegraphics[width=40mm]{engImages/deltaCycle.pdf} \caption{The cycle $C'$ and the tree $T$} \label{fig:deltaCycle} \end{center}\end{figure}Next, assume that there exists a cycle $C_1$ formed by $\stackrel\frown{v_j a}$ such that none of the vertices of $C_1$ is adjacent to $V(G) \setminus V(P_1)$. Choose the smallest cycle with this property. Obviously, this cycle is chordless. Now, three cases can be considered:\\\begin{enumerate}[leftmargin=5pt,label=(\roman*)]\item There exists a cycle $C_2$ formed by $P_2$ or $P_3$. Define the partition $C_1$, $C_2$ and a tree with the following edge set,$$E\Big(\langle V(G) \setminus \big(V(C_1) \cup V(C_2)\big)\rangle \Big) \bigcap \Big( \bigcup_{i=1}^3 E(P_i) \Big),$$and apply \hyperref[lemma:1]{Lemma 1}.\\\item There is no chordless cycle formed by $P_2$ and by $P_3$, and there is at least one edge between $V(P_2)$ and $V(P_3)$. Let $ab \in E(G)$, $a \in V(P_2)$ and $b \in V(P_3)$ and moreover $d_{P_2}(v_j, a) + d_{P_3}(v_j,b)$ is minimum. Notice that the cycle $\stackrel\frown{v_j a} \stackrel\frown{b v_j}$ is chordless. Let us call this cycle by $C_2$. Now, define the partition $C_2$ and a tree with the following edge set,$$E\Big(\langle V(G) \setminus V(C_2)\rangle \Big) \bigcap \Big( \bigcup_{i=1}^3 E(P_i) \Big),$$and apply \hyperref[lemma:1]{Lemma 1}.\\\item There is no chordless cycle formed by $P_2$ and by $P_3$, and there is no edge between $V(P_2)$ and $V(P_3)$. Let $C_2$ be the cycle consisting of two paths $P_2$ and $P_3$. Define the partition $C_2$ and a tree with the following edge set,$$E\Big(\langle V(G) \setminus V(C_2)\rangle \Big) \bigcap \Big( \bigcup_{i=1}^3 E(P_i) \Big),$$and apply \hyperref[lemma:1]{Lemma 1}.\end{enumerate}\end{enumerate}\vspace{5mm}\item\textbf{Case 2.}\label{case:2}Assume that $j < i$ for all Hamiltonian paths. Among all Hamiltonian paths consider the path such that $i'-j'$ is maximum. Now, three cases can be considered:\\\begin{enumerate}[leftmargin=0pt,label=]\item \textbf{Subcase 1.} There is no $s < j'$ and $t > i'$ such that $v_s v_t \in E(G)$. By \hyperref[remark:1]{Remark 1} $\,$ there are two chordless cycles $C_1$ and $C_2$, respectively formed by the paths $v_1 v_{j'}$ and $v_{i'} v_n$. By assumption there is no edge $xy$, where $x \in V(C_1)$ and $y \in V(C_2)$.Define a tree $T$ with the edge set:$$ E\Big(\langle V(G) \setminus \big(V(C_1) \cup V(C_2)\big) \rangle \Big) \bigcap \Big( E(P) \cup \{v_{i'}v_n, v_{j'}v_1\} \Big).$$Now, apply \hyperref[lemma:1]{Lemma 1} $\,$for the partition $\{T, C_1, C_2\}$.\\\item \textbf{Subcase 2.}\label{subcase:22} There are at least four indices $s, s' < j$ and $t, t' > i$ such that $v_s v_t, v_{s'} v_{t'} \in E(G)$. Choose four indices $g, h < j$ and $e, f > i$ such that $v_h v_e, v_g v_f \in E(G)$ and $|g-h| + |e-f|$ is minimum.\begin{figure}[H] \begin{center} \includegraphics[width=90mm]{engImages/case2-subcase2.pdf} \caption{Two edges $v_h v_e$ and $v_g v_f$} \label{fig:non-overlapping} \end{center}\end{figure}Three cases can be considered:\\\begin{enumerate}[leftmargin=0pt,label=(\alph*)]\item There is no chordless cycle formed by $\stackrel\frown{v_g v_h}$ and by $\stackrel\frown{v_e v_f}$.Consider the cycle $\stackrel\frown{v_g v_h} \stackrel\frown{v_e v_f}v_g$ and call it $C$. Now, define a tree $T$ with the edge set,$$\,\,\,E\Big(\langle V(G) \setminus V(C)\rangle \Big) \bigcap \Big( E(P) \cup \{v_1v_{j}, v_{i}v_n\} \Big),$$apply \hyperref[lemma:1]{Lemma 1} $\,$for the partition $\{T, C\}$.\\\item With no loss of generality, there exists a chordless cycle formed by $\stackrel\frown{v_e v_f}$ and there is no chordless cycle formed by the path $\stackrel\frown{v_g v_h}$. First suppose that there is a chordless cycle $C_1$ formed by $\stackrel\frown{v_e v_f}$ such that there is no edge between $V(C_1)$ and $\{v_1, \dots, v_j\}$. By \hyperref[remark:1]{Remark 1} $,$ there exists a chordless cycle $C_2$ formed by $\stackrel\frown{v_1 v_j}$. By assumption there is no edge between $V(C_1)$ and $V(C_2)$. Now, define a tree $T$ with the edge set,$$\quad\quad\quad\quad E\Big(\langle V(G) \setminus \big(V(C_1) \cup V(C_2)\big)\rangle \Big) \bigcap \Big( E(P) \cup \{v_1v_{j}, v_{i}v_n\} \Big),$$and apply \hyperref[lemma:1]{Lemma 1} $\,$for the partition $\{T, C_1, C_2\}$.$\;$ Next assume that for every cycle $C_r$ formed by $\stackrel\frown{v_e v_f}$, there are two vertices $x_r \in V(C_r)$ and $y_r \in \{v_1, \dots, v_j\}$ such that $x_r y_r \in E(G)$. Let $v_e=w_0, w_1, \dots, w_l=v_f$ be all vertices of the path $\stackrel\frown{v_e v_f}$ in $P$. Choose the shortest path $w_0 w_{i_1} w_{i_2} \dots w_l$ such that $0 < i_1 < i_2 < \dots < l$. Consider the cycle $w_0 w_{i_1} \dots w_l \stackrel\frown{v_g v_h}$ and call it $C$. Now, by removing $C$, $q$ vertex disjoint paths $Q_1, \dots, Q_q$ which are contained in $\stackrel\frown{v_e v_f}$ remain. Note that there exists a path of order $2$ in $C$ which by adding this path to $Q_i$ we find a cycle $C_{r_i}$, for some $i$. Hence there exists an edge $x_{r_i} y_{r_i}$ connecting $Q_i$ to $V(G) \setminus V(\stackrel\frown{v_e v_f})$. We define a tree $T$ whose edge set is the edges,$$\quad\quad\quad\quad\quad\quad E\Big(\langle V(G) \setminus V(C)\rangle \Big) \bigcap \Big( E(P) \cup \{v_1v_{j}, v_{i}v_n\} \cup \big\{x_{r_i} y_{r_i} \mid 1 \leq i \leq q\big\} \Big),$$then apply \hyperref[lemma:1]{Lemma 1} $\,$ on the partition $\{T, C\}$.\\\begin{figure}[H] \begin{center} \includegraphics[width=90mm]{engImages/deltaNonOverlapping.pdf} \caption{The tree $T$ and the shortest path $w_0 w_{i_1}\dots w_l$} \label{fig:delta-non-overlapping} \end{center}\end{figure}\item There are at least two chordless cycles, say $C_1$ and $C_2$ formed by the paths $\stackrel\frown{v_g v_h}$ and $\stackrel\frown{v_e v_f}$, respectively. Since $|g-h| + |e-f|$ is minimum, there is no edge $xy \in E(G)$ with $x \in V(C_1)$ and $y \in V(C_2)$. Now, define a tree $T$ with the edge set,$$\quad\quad\quad\quad E\Big( \langle V(G) \setminus \big(V(C_1) \cup V(C_2)\big) \rangle \Big) \bigcap \Big( E(P) \cup \{v_1 v_{j}, v_{i}v_n\} \Big),$$and apply \hyperref[lemma:1]{Lemma 1} $\,$for the partition $\{T, C_1, C_2\}$.\\\end{enumerate}\item \textbf{Subcase 3.} There exist exactly two indices $s,t$, $s < j' < i' < t$ such that $v_s v_t \in E(G)$ and there are no two other indices $s', t'$ such that $s' < j < i < t'$ and $v_{s'} v_{t'} \in E(G)$. We can assume that there is no cycle formed by $\stackrel\frown{v_{s+1} v_j}$ or $\stackrel\frown{v_i v_{t-1}}$, to see this by symmetry consider a cycle $C$ formed by $\stackrel\frown{v_{s+1} v_j}$. By \hyperref[remark:1]{Remark 1} $\,$ there exist chordless cycles $C_1$ formed by $\stackrel\frown{v_{s+1} v_j}$ and $C_2$ formed by $\stackrel\frown{v_{i} v_n}$. By assumption $v_s v_t$ is the only edge such that $s < j$ and $t > i \;$. Therefore, there is no edge between $V(C_1)$ and $V(C_2)$. Now, let $T$ be a tree defined by the edge set,$$ E\Big(\langle V(G) \setminus \big(V(C_1) \cup V(C_2)\big)\rangle \Big) \bigcap \Big( E(P) \cup \{v_1v_{j}, v_{i}v_n\} \Big),$$and apply \hyperref[lemma:1]{Lemma 1} $\,$for the partition \{$T$, $C_1$, $C_2$\}.\\$\quad$Furthermore, we can also assume that either $s \neq j'-1$ or $t \neq i'+1$, otherwise we have the Hamiltonian cycle $\stackrel\frown{v_1 v_s} \stackrel\frown{v_t v_n} \stackrel\frown{v_{i'} v_{j'}} v_1$ and by \cite[Theorem 9]{akbari} Conjecture \theconjecture$\,$ holds.$\quad$By symmetry, suppose that $s \neq j'-1$. Let $v_k$ be the vertex adjacent to $v_{j'-1}$, and $k \notin \{j'-2, j'\}$. It can be shown that $k > j'-1$, since otherwise by considering the Hamiltonian path $P': \; \stackrel\frown{ v_{k+1} v_{j'-1}}\stackrel\frown{v_k v_1} \stackrel\frown{v_{j'} v_n}$, the new $i'-j'$ is greater than the old one and this contradicts our assumption about $P$ in the \hyperref[case:2]{Case 2}.$\quad$We know that $j' < k < i$. Moreover, the fact that $\stackrel\frown{v_{s+1} v_j}$ does not form a cycle contradicts the case that $j' < k \le j$. So $j < k < i$. Consider two cycles $C_1$ and $C_2$, respectively with the vertices $v_1 \stackrel\frown{v_{j'} v_{j}} v_1$ and $v_n \stackrel\frown{v_{i'} v_{i}} v_n$. The cycles $C_1$ and $C_2$ are chordless, otherwise there exist cycles formed by the paths $\stackrel\frown{v_{s+1} v_j}$ or $\stackrel\frown{v_i v_{t-1}}$. Now, define a tree $T$ with the edge set$$ E\Big(\langle V(G) \setminus \big(V(C_1) \cup V(C_2)\big)\rangle \Big) \bigcap \Big( E(P) \cup \{v_s v_t, v_k v_{j'-1}\} \Big),$$and apply \hyperref[lemma:1]{Lemma 1} $\,$for the partition \{$T$, $C_1$, $C_2$\}.\end{enumerate}\end{enumerate}\end{proof}\noindent\textbf{Remark 2.}\label{remark:2}Indeed, in the proof of the previous theorem we showed a stronger result, that is, for every traceable cubic graph there is a decomposition with at most two cycles.

{'timestamp': '2016-07-19T02:04:55', 'yymm': '1607', 'arxiv_id': '1607.04768', 'language': 'en', 'url': 'https://arxiv.org/abs/1607.04768'}

arxiv

\section{Principle of nano strain-amplifier}\begin{figure*}[t!]\centering\includegraphics[width=5.4in]{Fig1}\vspace{-0.5em}\caption{Schematic sketches of nanowire strain sensors. (a)(b) Conventional non-released and released NW structure; (c)(d) The proposed nano strain-amplifier and its simplified physical model.}\label{fig:fig1}\vspace{-1em}\end{figure*}Figure \ref{fig:fig1}(a) and 1(b) show the concept of the conventional structures of piezoresistive sensors. The piezoresistive elements are either released from, or kept on, the substrate. The sensitivity ($S$) of the sensors is defined based on the ratio of the relative resistance change ($\Delta R/R$) of the sensing element and the strain applied to the substrate ($\varepsilon_{sub}$):\begin{equation}S = (\Delta R/R)/\varepsilon_{sub}\label{eq:sensitivity}\end{equation}In addition, the relative resistance change $\Delta R/R$ can be calculated from the gauge factor ($GF$) of the material used to make the piezoresistive elements: $\Delta R/R = GF \varepsilon_{ind}$, where $\varepsilon_{ind}$ is the strain induced into the piezoresistor. In most of the conventional strain gauges as shown in Fig. \ref{fig:fig1} (a,b), the thickness of the sensing layer is typically below a few hundred nanometers, which is much smaller than that of the substrate. Therefore, the strain induced into the piezoresistive elements is approximately the same as that of the substrate ($\varepsilon_{ind} \approx \varepsilon_{sub}$). Consequently, to improve the sensitivity of strain sensors (e.g. enlarging $\Delta R/R$), electrical approaches which can enlarge the gauge factor ($GF$) are required. Nevertheless, as aforementioned, the existence of the large gauge factor in nanowires due to quantum confinement or surface state, is still considered as controversial. It is also evident from Eq. \ref{eq:sensitivity} that the sensitivity of strain sensors can also be improved using a mechanical approach, which enlarges the strain induced into the piezoresistive element. Figure \ref{fig:fig1}(c) shows our proposed nano strain-amplifier structure, in which the piezoresistive nanowires are locally fabricated at the centre of a released bridge. The key idea of this structure is that, under a certain strain applied to the substrate, a large strain will be concentrated at the locally fabricated SiC nanowires. The working principle of the nano strain-amplifier is similar to that of the well-known dogbone structure, which is widely used to characterize the tensile strength of materials \cite{dogbone1,dogbone2}. That is, when a stress is applied to the dogbone-shape of a certain material, a crack, if generated, will occur at the middle part of the dogbone. The large strain concentrated at the narrow area located at the centre part with respect to the wider areas located at outer region, causes the crack. Qualitative and quantitative explanations of the nano strain-amplifier are presented as follows. For the sake of simplicity, the released micro frame and nanowire (single wire or array) of the nano strain-amplifier can be considered as solid springs, Fig. \ref{fig:fig1}(d). The stiffness of these springs are proportional to their width ($w$) and inversely proportional to their length (l): $K \propto w/l$. Consequently, the model of the released nanowire and micro frames can be simplified as a series of springs, where the springs with higher stiffness correspond to the micro frame, and the single spring with lower stiffness corresponds to the nanowire. It is well-known in classical physics that, for serially connected springs, a larger strain will be concentrated in the low--stiffness string, while a smaller strain will be induced in the high--stiffness string \cite{Springbook}. The following analysis quantitatively explained the amplification of the strain.\begin{figure}[b!]\centering\includegraphics[width=3in]{Fig2}\vspace{-1em}\caption{Finite element analysis of the strain induced in to the nanowire array utilizing nano strain-amplifier.}\label{fig:fig2}\end{figure}When a tensile mechanical strain ($\varepsilon_{sub}$) is applied to the substrate, the released structure will also be elongated. Since the stiffness of the released frame is much smaller than that of the substrate, it is safe to assume that the released structure will follows the elongation of the substrate. The displacement of the released structure $\Delta L$ is:\begin{equation}\Delta L = \Delta L_m + \Delta L_n = L_m \varepsilon_m + L_n \varepsilon_n\label{eq:displacement}\end{equation} where $L_m$, $L_n$ are the length; $\Delta L_m$, $\Delta L_n$ are the displacement; and $\varepsilon_m$, $\varepsilon_n$ are the strains induced into the micro spring and nano spring, respectively. The subscripts m and n stand for the micro frames and nanowires, respectively. Furthermore, due to the equilibrium of the stressing force ($F$) along the series of springs, the following relationship is established: $F= K_m\Delta L_m = K_n \Delta L_n$, where $K_m$, $K_n$ are the stiffness of the released micro frames and nanowires, respectively. Consequently the relationship between the displacement of the micro frame (higher stiffness) and nanowires (lower stiffness) is:\begin{equation}\frac{\Delta L_m}{\Delta L_n}=\frac{K_n}{K_m}=\frac{L_mw_n}{L_nw_m}\label{eq:euili}\end{equation}Substituting Eqn. \ref{eq:euili} into Eqn. \ref{eq:displacement}, the strain induced into the locally fabricated nanowires is:\begin{equation}\varepsilon_n = \frac{\Delta L_n}{L_n} = \frac{1}{1-\frac{w_m-w_n}{w_m}\frac{L_m}{L}}\varepsilon_{sub}\label{eq:strainamp}\end{equation} Equation \ref{eq:strainamp} indicates that increasing the ratio of $w_m/w_n$ and $L_m/L_n$ significantly amplifies the strain induced into the nanowire from the strain applied to the substrate. This model is also applicable to the case of nanowire arrays, in which $w_n$ is the total width of all nanowires in the array.The theoretical model is then verified using the finite element analysis (FEA). In the FEA simulation, we compare the strain induced into (i) non released nanowires, (ii) the conventionally released nanowires, and (iii) our nano strain-amplifier structure, using COMSOL Multiphysics \texttrademark. In our nano strain amplifying structure, the width of the released frame was set to be 8 $\mu$m, while the width of each nanowire in the array (3 wires) was set to be 370 nm. The nanowires array structure was selected as it can enhance the electrical conductance of the SiC nanowires resistor which makes the subsequent experimental demonstration easier. The ratio between the length of nanowires and micro bridge was set to be 1: 20. With this geometrical dimensions, strain induced into nanowires array $\varepsilon_n$ was numerically calculated to be approximately 6 times larger than $\varepsilon_{sub}$, Eqn. \ref{eq:strainamp}. The simulation results show that for all structure, the elongation of non-released and released nanowires follow that of the substrate. In addition, strain was almost completely transferred into conventional released and non-released structures. Furthermore, the ratio of the strain induced in to the locally fabricated nanowires was estimated to be 5.9 times larger than that of the substrate, Fig. \ref{fig:fig2}. These results are in solid agreement with the theoretical analysis presented above. For a nanowire array with an average width of 470 nm, the amplified gain of strain was found to be 4.5. Based on the theoretical analysis, we conducted the following experiments to demonstrate the high sensitivity of SiC nanowire strain sensors using the nano strain-amplifier. A thin 3C-SiC film with its thickness of 300 nm was epitaxially grown on a 150 mm diameter Si wafer using low pressure chemical vapour deposition \cite{SiC_growth}. The film was \emph{in situ} doped using Al dopants. The carrier concentration of the p-type 3C-SiC was found to be $5 \times 10^{18}$ cm$^{-3}$, using a hot probe technique \cite{philip}. The details of the characteristics of the grown film can be found elsewhere \cite{Phan_JMC}. Subsequently, I-shape p-type SiC resistors with aluminum electrodes deposited on the surface were patterned using inductive coupled plasma (ICP) etching. As the piezoresistance of p-type 3C-SiC depends on crystallographic orientation, all SiC resistors of the present work were aligned along [110] direction to maximize the piezoresistive effect. Next, the micro scale SiC resistors were then released from the Si substrate using dry etching (XeF$_2$). Finally, SiC nanowire arrays were formed at the centre of the released bridge using focused ion beam (FIB). Two types of nanowire array were fabricated with three nanowires for each array. The average width of each nanowire in each type were 380 nm and 470 nm, respectively. Figure \ref{fig:fig3} shows the SEM images of the fabricated samples, including the conventional released structure, non-released nanowires, and the nano strain-amplifier. \begin{figure}[t!]\centering\includegraphics[width=3in]{Fig3}\caption{SEM image of SiC strain sensors. (a) Released SiC micro bridge used for the subsequent fabrication of the nano strain-amplifier; (b) SEM of a micro SiC resistor where the SiC nanowires array were formed using FIB; (c) SEM of non-released SiC nanowires; (d) SEM of locally fabricated SiC nanowires released from the Si substrate (nano strain-amplifier).}\label{fig:fig3}\vspace{-1em}\end{figure}The current voltage (I-V) curves of all fabricated samples were characterized using a HP 4145 \texttrademark ~parameter analyzer. The linear relationship between the applied voltage and measured current, indicated that Al made a good Ohmic contact with the highly doped SiC resistance, Fig. \ref{fig:IV}. Additionally, the electrical conductivity of both nanowires and micro frame estimated from the I-V curve and the dimensions of the resistors shows almost the same value. This indicated that the FIB process did not cause a significant surface damage to the fabricated nanowires. \begin{figure}[b!]\centering\includegraphics[width=3in]{Fig4}\vspace{-1.5em}\caption{Current voltage curves of the fabricated SiC resistors.}\label{fig:IV}\end{figure}The bending experiment was used to characterize the piezoresistive effect in micro size SiC resistors and locally fabricated SiC nanowire array. In this experiment one end of the Si cantilever (with a thickness of 625 $\mu$m, and a width of 7 mm) was fixed while the other end was deflected by applying different forces. The distance from the fabricated nanowires to the free end of the Si cantilever was approximately 45 mm. The strain induced into the Si substrate is $\varepsilon_\text{sub} = Mt/2EI$, where $M$ is the applied bending moment; and $t$, $E$ and $I$ are the thickness, Young's modulus and the moment of inertia of the Si cantilever, respectively. The response of the SiC resistance to applied strain was then measured using a multimeter (Agilent \texttrademark 34401 A).\begin{figure}[h!]\centering\includegraphics[width=3in]{Fig5.eps}\vspace{-1.5em}\caption{Experimental results. (a) A comparision between the relative resistance change in the nano strain-amplifiers, non released nanowires and released micro frames; (b) The repeatability of the SiC nanowires strain sensors utilizing the proposed structure.}\label{fig:DRR}\vspace{-1em}\end{figure}The relative resistance change ($\Delta R/R$) of the micro and nano SiC resistors was plotted against the strain induced into the Si substrate $\varepsilon_{sub}$, Fig. \ref{fig:DRR}(a). For all fabricated samples, the relative resistance change shows a good linear relationship with the applied strain ($\varepsilon_{sub}$). In addition, with the same applied strain to the Si substrate, the resistance change of the SiC nanowires using the nano strain-amplifier was much larger than that of the the SiC micro resistor and the conventional non-released SiC nanowires. In addition, reducing the width of the SiC nanowires also resulted in the increase of the sensitivity. The magnitude of the piezoresistive effect in the nano strain-amplifier as well as conventional structures were then quantitatively evaluated based on the effective gauge factor ($GF_{eff}$), which is defined as the ratio of the relative resistance change to the applied strain to the substrate: $GF_{eff} = (\Delta R/R)/\varepsilon_{sub}$. Accordingly, the effective gauge factor of the released micro SiC was found to be 28, while that of the non-released SiC nanowires was 35. From the data shown in Fig. \ref{fig:DRR}, the effective gauge factor of the 380 nm and 470 nm SiC nanowires in the nano strain-amplifier were calculated as 150 and 124, respectively. Thus for nanowire arrays with average widths of 380 nm and 470 nm, the sensitivity of the nano strain-amplifier was 5.4 times and 4.6 times larger than the bulk SiC, respectively. These results were consistent with analytical and numerical models presented above. The relative resistance change of the nano strain-amplifier also showed excellent linearity with the applied strain, with a linear regression of above 99\%. The resistance change of the nano strain-amplifier can also be converted into voltage signals using a Wheatstone bridge, Fig. \ref{fig:DRR}(b). The output voltage of the nano strain-amplifier increases with increasing tensile strains from 0 ppm to 180 ppm, and returned to the initial value when the strain was completely removed, confirming a good repeatability after several strain induced cycles. The linearity of the relative resistance change, and the repeatability indicate that the proposed structure is promising for strain sensing applications. In conclusion, this work presents a novel mechanical approach to obtain highly sensitive piezoresistance in nanowires based on a nano strain-amplifier. The key factor of the nano strain-amplifier lies on nanowires locally fabricated on a released micro structure. Experimental studies were conducted on SiC nanowires, confirming that by utilizing our nano strain-amplifier, the sensitivity of SiC nanowires was 5.4 times larger than that of conventional structures. This result indicated that the nano strain-amplifier is an excellent platform for ultra sensitive strain sensing applications.

{'timestamp': '2016-07-18T02:07:38', 'yymm': '1607', 'arxiv_id': '1607.04531', 'language': 'en', 'url': 'https://arxiv.org/abs/1607.04531'}

arxiv

\section{Introduction}\label{intro}Gas has a fundamental role in shaping the evolution of galaxies,through its accretion on to massive haloes, cooling and subsequentfuelling of star formation, to the triggering of extreme luminousactivity around super massive black holes. Determining how thephysical state of gas in galaxies changes as a function of redshift istherefore crucial to understanding how these processes evolve overcosmological time. The standard model of the gaseous interstellarmedium (ISM) in galaxies comprises a thermally bistable medium(\citealt*{Field:1969}) of dense ($n \sim 100$\,cm$^{-3}$) coldneutral medium (CNM) structures, with kinetic temperatures of $T_{\rm k} \sim 100$\,K, embedded within a lower-density ($n \sim1$\,cm$^{-3}$) warm neutral medium (WNM) with $T_{\rm k} \sim10^{4}$\,K. The WNM shields the cold gas and is in turn ionized bybackground cosmic rays and soft X-rays (e.g. \citealt{Wolfire:1995, Wolfire:2003}). A further hot ($T_{\rm k} \sim 10^{6}$\,K) ionizedcomponent was introduced into the model by \cite{McKee:1977}, toaccount for heating by supernova-driven shocks within the inter-cloudmedium. In the local Universe, this paradigm has successfullywithstood decades of observational scrutiny, although there is someevidence (e.g. \citealt{Heiles:2003b}; \citealt*{Roy:2013b};\citealt{Murray:2015}) that a significant fraction of the WNM mayexist at temperatures lower than expected for global conditions ofstability, requiring additional dynamical processes to maintain localthermodynamic equilibrium.Since atomic hydrogen (\mbox{H\,{\sc i}}) is one of the most abundantcomponents of the neutral ISM and readily detectable through eitherthe 21\,cm or Lyman $\alpha$ lines, it is often used as a tracer ofthe large-scale distribution and physical state of neutral gas ingalaxies. The 21\,cm line has successfully been employed in surveyingthe neutral ISM in the Milky Way(e.g. \citealt{McClure-Griffiths:2009,Murray:2015}), the Local Group(e.g. \citealt{Kim:2003,Bruns:2005,Braun:2009,Gratier:2010}) andlow-redshift Universe (see \citealt{Giovanelli:2016} for areview). However, beyond $z \sim 0.4$ (\citealt{Fernandez:2016})\mbox{H\,{\sc i}} emission from individual galaxies becomes too faintto be detectable by current 21\,cm surveys and so we must rely onabsorption against suitably bright background radio (21\,cm) or UV(Lyman-$\alpha$) continuum sources to probe the cosmological evolutionof \mbox{H\,{\sc i}}. The bulk of neutral gas is contained inhigh-column-density damped Lyman-$\alpha$ absorbers (DLAs, $N_{\rm HI}\geq 2 \times 10^{20}$\,cm$^{-2}$; see \citealt*{Wolfe:2005} for areview), which at $z \gtrsim 1.7$ are detectable in the opticalspectra of quasars. Studies of DLAs provide evidence that the atomicgas in the distant Universe appears to be consistent with amulti-phase neutral ISM similar to that seen in the Local Group(e.g. \citealt*{Lane:2000}; \citealt*{Kanekar:2001c};\citealt*{Wolfe:2003b}). However, there is some variation in the coldand warm fractions measured throughout the DLA population(e.g. \citealt*{Howk:2005}; \citealt{Srianand:2005, Lehner:2008};\citealt*{Jorgenson:2010}; \citealt{Carswell:2011, Carswell:2012, Kanekar:2014a}; \citealt*{Cooke:2015}; \citealt*{Neeleman:2015}).The 21-cm spin temperature affords us an important line-of-enquiry inunraveling the physical state of high-redshift atomic gas. Thisquantity is sensitive to the processes that excite the ground-state of\mbox{H\,{\sc i}} in the ISM(\citealt{Purcell:1956,Field:1958,Field:1959b,Bahcall:1969}) andtherefore dictates the detectability of the 21\,cm line in absorption.In the CNM the spin temperature is governed by collisional excitationand so is driven to the kinetic temperature, while the lower densitiesin the WNM mean that the 21\,cm transition is not thermalized bycollisions between the hydrogen atoms, and so photo-excitation by thebackground Ly $\alpha$ radiation field becomes important. Consequentlythe spin temperature in the WNM is lower than the kinetic temperature,in the range $\sim$1000 -- 5000\,K depending on the column density andnumber of multi-phase components (\citealt{Liszt:2001}). Importantly,the spin temperature measured from a single detection of extragalacticabsorption is equal to the harmonic mean of the spin temperature inindividual gas components, weighted by their column densities, therebyproviding a method of inferring the CNM fraction in high-redshiftsystems.Surveys for 21\,cm absorption in known redshifted DLAs have been usedto simultaneously measure the column density and spin temperature of\mbox{H\,{\sc i}} (see \citealt{Kanekar:2014a} and referencestherein). There is some evidence for an increase (at $4\,\sigma$significance) in the spin temperature of DLAs at redshifts above $z =2.4$, and a difference (at $6\,\sigma$ significance) between thedistribution of spin temperatures in DLAs and the Milky Way(\citealt{Kanekar:2014a}). The implication that at least 90\,per\,centof high-redshift DLAs may have CNM fractions significantly less thanthat measured for the Milky Way has important consequences for theheating and cooling of neutral gas in the early Universe and starformation (e.g. \citealt*{Wolfe:2003a}). However, these targetedobservations rely on the limited availability of simultaneous 21\,cmand optical/UV data for the DLAs and assumes commonality between thecolumn density probed by the optical and radio sight-lines. The firstissue can be overcome by improving the sample statistics throughlarger 21\,cm line surveys of high-redshift DLAs, but the latterrequires improvements to our methodology and understanding of the gasdistribution in these systems. There are also concerns about theaccuracy to which the fraction of the source structure subtended bythe absorber can be measured in each system, which can only beresolved through spectroscopic very long baseline interferometry(VLBI). It has been suggested that the observed evolution in spintemperature could be biased by assumptions about the radio-sourcecovering factor (\citealt{Curran:2005}) and its behaviour as afunction of redshift (\citealt{Curran:2006b, Curran:2012b}).In this paper we consider an approach using the statistical constrainton the average spin temperature achievable with future large 21\,cmsurveys using precursor telescopes to the Square Kilometre Array(SKA). This will enable independent verification of the evolution inspin temperature at high redshift and provide a method of studying theglobal properties of neutral gas below $z \approx 1.7$, where theLyman\,$\alpha$ line is inaccessible using ground-based observatories.In an early attempt at a genuinely blind 21\,cm absorption survey,\cite{Darling:2011} used pilot data from the Arecibo Legacy FastArecibo L-band Feed Array (ALFALFA) survey to obtain upper limits onthe column density frequency distribution from 21\,cm absorption atlow redshift ($z \lesssim 0.06$). However, they also noted that thenumber of detections could be used to make inferences about the ratioof the spin temperature to covering factor. Building upon this work,\cite{Wu:2015} found that their upper limits on the frequencydistribution function measured from the 40\,per\,cent ALFALFA survey({$\alpha$}.40; \citealt{Haynes:2011}) could only be reconciled withmeasurements from other low-redshift 21\,cm surveys if the typicalspin temperature to covering factor ratio was greater than 500\,K. Athigher redshifts, \cite{Gupta:2009} found that the number density of21\,cm absorbers in known \mbox{Mg\,{\sc ii}} absorbers appeared todecrease with redshift above $z \sim 1$, consistent with a reductionin the CNM fraction. We pursue this idea further by investigatingwhether future wide-field 21\,cm surveys can be used to measure theaverage spin temperature in distant galaxies that are rich in atomicgas.\section{The expected number of intervening \mbox{H\,{\sc i}} absorbers}\label{section:expected_number}We estimate the expected number of intervening \mbox{H\,{\sc i}}systems towards a sample of background radio sources by evaluating thefollowing integral over all sight-lines\begin{equation}\label{equation:expected_number} \mu = \iint{f(N_{\rm HI},X)\,\mathrm{d}X\,\mathrm{d}N_{\rm HI}}, \end{equation}where $f(N_{\rm HI}, X)$ is the frequency distribution as a functionof column density ($N_{\rm HI}$) and comoving path length ($X$). Weuse the results of recent surveys for 21\,cm emission in nearbygalaxies (e.g. \citealt{Zwaan:2005}) and high-redshift Lyman-$\alpha$absorption in the Sloan Digitial Sky Survey (SDSS;e.g. \citealt*{Prochaska:2005}; \citealt{Noterdaeme:2009}), which showthat $f(N_{\rm HI}, X)$ can be parametrized by a gamma function of theform\begin{equation} f(N_{\rm HI}, X) = \left({f_{\ast} \over N_{\ast}}\right)\left({N_{\rm HI} \over N_{\ast}}\right)^{-\beta}\exp{\left(-{N_{\rm HI} \over N_{\ast}}\right)}\,\mathrm{cm}^{2},\end{equation}where $f_{\ast} = 0.0193$, $\log_{10}(N_{\ast}) = 21.2$ and $\beta =1.24$ at $z = 0$ (\citealt{Zwaan:2005}), and $f_{\ast} = 0.0324$,$\log_{10}(N_{\ast}) = 21.26$ and $\beta = 1.27$ at $z \approx 3$(\citealt{Noterdaeme:2009}). While the observational data do not yetconstrain models for evolution of the \mbox{H\,{\sc i}} distributionat intermediate redshifts between $z \sim 0.1$ and$3$\footnote{Measurements of $f(N_{\rm HI},X)$ at intermediate redshifts come from targeted ultra-violet surveys of DLAs using the \emph{Hubble Space Telescope} (\citealt*{Rao:2006}; \citealt{Neeleman:2016}). However, due to the limited sample sizes these are currently an order-of-magnitude less sensitive than the nearby 21-cm and high-redshift optical Lyman-$\alpha$ surveys.}, itis known to be much weaker than the significant decline seen in theglobal star-formation rate and molecular gas over the same epoch(e.g. \citealt{Lagos:2014}). We therefore carry out a simple linearinterpolation between the low and high redshift epochs to estimate$f(N_{\rm HI},X)$ as a function of redshift.\begin{figure}\centering\includegraphics[width=0.475\textwidth]{width_dist.pdf}\caption{The distribution of 21\,cm line widths based on existing detections of intervening absorption at $z > 0.1$ (see the text for details of this sample). The sample size in each bin is denoted by the number above and errorbars denote the standard deviation. The solid red line is a log-normal fit to the data, from which we draw random samples for our analysis.}\label{figure:width_dist}\end{figure}The probability of detecting an absorbing system of given columndensity depends on the sensitivity of the survey, the flux density andstructure of the background source and the fraction of \mbox{H\,{\sc i}} in the lower spin state, given by the spin temperature. Weexpress the column density ($N_{\rm HI}$; in atoms\,cm$^{-2}$) interms of the optical depth ($\tau$) and spin temperature ($T_{\rm spin}$; in K) by\begin{equation}\label{equation:column_density} N_{\rm HI} = 1.823\times10^{18}\,T_{\rm spin} \int{\tau(v)\mathrm{d}v},\end{equation}where the integral is performed across the spectral line in the systemrest-frame velocity $v$ (in km\,s$^{-1}$). We then express the opticaldepth in terms of the observables as\begin{equation} \tau = -\ln\left[1 + {\Delta{S}\over c_{\rm f}S_{\rm cont}}\right],\end{equation}where $\Delta{S}$ is the observed change in flux density due toabsorption, $S_{\rm cont}$ is the background continuum flux densityand $c_{\rm f}$ is the (often unknown) fraction of background fluxdensity subtended by the intervening gas.\begin{figure}\centering\includegraphics[width=0.465\textwidth]{covfact_dist.pdf}\caption{The distribution of \mbox{H\,{\sc i}} covering factors from the main sample of \citet{Kanekar:2014a}, which were estimated using the fraction of total continuum flux density in the quasar core. The sample size in each bin is denoted by the number above and errorbars denote the standard deviation. The solid red line is the uniform distribution, from which we draw random samples for our analysis.}\label{figure:covfact_dist}\end{figure}We assume that a single intervening system can be described by aGaussian velocity distribution of full width at half maximum (FWHM)dispersion ($\Delta{v_{\rm 50}}$) and peak optical depth ($\tau_{\rm peak}$), so that \autoref{equation:column_density} can be re-writtenas\begin{equation}\label{equation:column_density_gaussian} N_{\rm HI} = 1.941\times10^{18}\,T_{\rm spin}\,\tau_{\rm peak}\,\Delta{v_{\rm 50}}.\end{equation}If we further assume that the rms spectral noise is Gaussian, with astandard deviation $\sigma_{\rm chan}$ per independent channel$\Delta{v_{\rm chan}}$, then the 5$\sigma$ column density detectionlimit is given by\begin{equation} N_{5\sigma} \approx 1.941\times10^{18}\,T_{\rm spin}\,\tau_{\rm 5\sigma}\,\Delta{v_{\rm conv}},\end{equation}where \begin{equation}\label{equation:optical_depth_limit} \tau_{5\sigma} \approx -\ln\left[1 - {5\,\sigma_{\rm chan}\over c_{\rm f}\,S_{\rm cont}}\sqrt{\Delta{v}_{\rm chan}\over \Delta{v_{\rm conv}}}\right],\end{equation}and $\Delta{v_{\rm conv}} \approx \sqrt{\Delta{v}_{\rm chan}^{2} + \Delta{v}_{50}^{2}}$, which is the observed width of the line, givenby the convolution of the physical velocity distribution and thespectral resolution of the telescope. We now redefine $\mu$ as theexpected number of intervening \mbox{H\,{\sc i}} detections in oursurvey as a function of the column density sensitivity along eachsight-line where each comoving path element$\delta{X}(z)$\footnote{For the purposes of this work we adopt a flat $\Lambda$ cold dark matter cosmology with $H_{0}$ = 70\,km\,s$^{-1}$, $\Omega_\mathrm{M}$ = 0.3 and $\Omega_{\Lambda}$ = 0.7. } in the integral defined by\autoref{equation:expected_number} is given by\begin{equation} \delta{X}(z)= \begin{cases} {\delta{z}\,(1+z)^{2}\over \sqrt{(1+z)^{2}(1+z\Omega_{\rm M})-z(z+2)\Omega_{\rm \Lambda}}}, & \text{if}\ N_{\rm HI} \geq N_{5\sigma}, \\ 0, & \text{otherwise}.\end{cases}\end{equation}To calculate the column density sensitivity for each comoving elementwe draw random samples for $\Delta{v}_{50}$ and $c_{\rm f}$ fromcontinuous prior distributions based on existing evidence. In the caseof $\Delta{v}_{50}$ we use a log-normal distribution obtained from asimple least-squares fit to the sample distribution from previous21-cm absorption surveys reported in the literature (see\autoref{figure:width_dist})\footnote{References for the literature sample of line widths shown in \autoref{figure:width_dist}: \citet*{Briggs:2001}; \citet*{Carilli:1993}; \citet*{Chengalur:1999}; \citet{Chengalur:2000, Curran:2007b, Davis:1978, Ellison:2012, Gupta:2009, Gupta:2012, Gupta:2013}; \citet{Kanekar:2001b,Kanekar:2003b}; \citet{Kanekar:2001c, Kanekar:2006, Kanekar:2009a, Kanekar:2013, Kanekar:2014a}; \citet{Kanekar:2003a}, \citet*{Kanekar:2007}; \citet*{Kanekar:2014b}; \citet{ Lane:2001, Lovell:1996, York:2007, Zwaan:2015}.}, assuming that this correctly describes the truedistribution for the population of DLAs. However, direct measurementof the \mbox{H\,{\sc i}} covering factor is significantly moredifficult and so for the purposes of this work we draw random samplesassuming a uniform distribution between 0 and 1. In\autoref{figure:covfact_dist}, we show a comparison between thisassumption and the sample distribution estimated by\cite{Kanekar:2014a} from their main sample of 37 quasars. Kanekar etal. used VLBI synthesis imaging to measure the fraction of totalquasar flux density contained within the core, which was then used asa proxy for the covering factor. By carrying out a two-tailedKolmogorov-Smirnov (KS) test of the hypothesis that the Kanekar etal. data are consistent with our assumed uniform distribution, we findthat this hypothesis is rejected at the 0.05 level, but not at the0.01 level (this outcome is dominated by the paucity of quasars in thesample with $c_{\rm f} \lesssim 0.2$). It is therefore possible thatthe population distribution of \mbox{H\,{\sc i}} covering factors maydeviate somewhat from the uniform distribution assumed in thiswork. We discuss the implications of this further in\autoref{section:covering_factor}.\section{A 21\,cm absorption survey with ASKAP}\label{section:all_sky_survey}We use the Australian Square Kilometre Array Pathfinder (ASKAP;\citealt{Johnston:2007}) as a case study to demonstrate the expectedresults from planned wide-field surveys for 21\,cm absorption(e.g. the ASKAP First Large Absorption Survey in \mbox{H\,{\sc i}} --Sadler et al., the MeerKAT Absorption Line Survey -- Gupta et al., andthe Search for HI absorption with AperTIF -- Morganti et al.). ASKAPis currently undergoing commissioning. Proof-of-concept observationswith the Boolardy Engineering Test Array (\citealt{Hotan:2014}) havealready been used to successfully detect a new \mbox{H\,{\sc i}}absorber associated with a probable young radio galaxy at $z = 0.44$(\citealt{Allison:2015a}). Here we predict the outcome of a future2\,h-per-pointing survey of the entire southern sky ($\delta \leq+10\degr$) using the full 36-antenna ASKAP in a single 304\,MHz bandbetween 711.5 and 1015.5\,MHz, equivalent to \mbox{H\,{\sc i}}redshifts between $z = 0.4$ and 1.0.\begin{figure}\centering\includegraphics[width=0.475\textwidth]{nsources_flux.pdf}\caption{The number of radio sources in our simulated southern sky survey ($\delta \leq +10\degr$) estimated from existing catalogues at $L$-band frequencies (see the text for details). The grey region encloses the expected number across the 711.5 - 1015.5\,MHz ASKAP frequency band, assuming a canonical spectral index of $\alpha = -0.7$.}\label{figure:nsources_flux}\end{figure}Our expectations of the ASKAP performance are based on preliminarymeasurements by \cite{Chippendale:2015} using the prototype Mark {\sc II} phase array feed. We estimate the noise per spectral channelusing the radiometer equation\begin{equation} \sigma_{\rm chan} = {S_{\rm system} \over \sqrt{n_{\rm pol}\,n_{\rm ant}\,(n_{\rm ant} - 1)\,\Delta{t}_{\rm in}\,\Delta{\nu}_{\rm chan}}}, \end{equation}where $S_{\rm system}$ is the system equivalent flux density, $n_{\rm pol}$ is the number of polarizations, $n_{\rm ant}$ is the number ofantennas, $\Delta{t}_{\rm in}$ is the on-source integration time and$\Delta{\nu}_{\rm chan}$ is the spectral resolution in frequency. Thesensitivity of the telescope in the 711.5 - 1015.5\,MHz band isexpected to vary between $S_{\rm system} \approx 3200$ and $2000$\,Jy,with the largest change in sensitivity between 700 and 800\,MHz. ASKAPhas dual linear polarization feeds, 36 antennas and a fine filter bankthat produces 16\,416 independent channels across the full 304\,MHzbandwidth, so the expected noise per 18.5\,kHz channel in a 2\,hobservation is approximately 5.5 - 3.5\,mJy\,beam$^{-1}$ across theband. In the case of an actual survey, the true sensitivity will ofcourse be recorded in the spectral data as a function of redshift (seee.g. \citealt{Allison:2015a}), but for the purposes of the simulatedsurvey presented in this work we split the band into several frequencybins to capture the variation in sensitivity and velocity resolution(which is in the range 7.8\,km\,s$^{-1}$ at 711.5\,MHz to5.5\,km\,s$^{-1}$ at 1015.5\,MHz).\begin{figure}\centering\includegraphics[width=0.475\textwidth]{zdist.pdf}\caption{The distribution of CENSORS sources (\citealt{Brookes:2008}) brighter than 10\,mJy beyond a given redshift $z$. The red line denotes the cumulative distribution calculated from the parametric model of \citet{deZotti:2010}.}\label{figure:zdist}\end{figure}In order to simulate a realistic survey of the southern sky we selectall radio sources south of $\delta = +10\degr$ from catalogues of theNational Radio Astronomy Observatory Very Large Array Sky Survey(NVSS, $\nu = 1.4$\,GHz, $S_{\rm src} \gtrsim 2.5$\,mJy;\citealt{Condon:1998}), the Sydney University Molonglo SkySurvey ($\nu= 843$\,MHz, $S_{\rm src} \gtrsim 10$\,mJy; \citealt{Mauch:2003}) andthe second epoch Molonglo Galactic Plane Survey ($\nu = 843$\,MHz,$S_{\rm src} \gtrsim 10$\,mJy; \citealt{Murphy:2007}). The source fluxdensities, used to calculate the optical depth limit in\autoref{equation:optical_depth_limit}, are estimated at the centre ofeach frequency bin by extrapolating from the catalogue values andassuming a canonical spectral index of $\alpha = -0.7$. In\autoref{figure:nsources_flux}, we show the resulting cumulativedistribution of radio sources in our sample as a function of fluxdensity across the band.\begin{figure}\centering\includegraphics[width=0.475\textwidth]{nsources_tau.pdf}\caption{The number of sources in our simulated ASKAP survey with a 21\,cm opacity sensitivity greater than or equal to $\tau_{5\sigma}$, as defined by \autoref{equation:optical_depth_limit}. The grey region encloses opacity sensitivities for the 711.5 - 1011.5\,MHz band. Random samples for the line FWHM and covering factor were drawn from the distributions shown in \autoref{figure:width_dist} and \autoref{figure:covfact_dist}.}\label{figure:nsources_tau}\end{figure} For any given sight-line, the redshift interval over which absorptionmay be detected is dependent upon the distance to the continuumsource. The lack of accurate spectroscopic redshift measurements formost radio sources over the sky necessitates the use of a statisticalapproach based on a model for the source redshift distribution. Wetherefore apply a statistical weighting to each comoving path element$\delta{X}(z)$ such that the expected number of absorber detections isnow given by\begin{equation}\label{equation:weighted_sum_number} \mu = \iint{f(N_{\rm HI},X)\,\mathcal{F}_{\rm src}(z^{\prime} \geq z)\,\mathrm{d}X\,\mathrm{d}N_{\rm HI}}, \end{equation}where\begin{equation} \mathcal{F}_{\rm src}(z^{\prime} \geq z) = {\int_{z}^{\infty} \mathcal{N}_{\rm src}(z^{\prime})\mathrm{d}z^{\prime}\over\int_{0}^{\infty}\mathcal{N}_{\rm src} (z^{\prime})\mathrm{d}z^{\prime}},\end{equation}and $\mathcal{N}_{\rm src}(z)$ is the number of radio sources as afunction of redshift. To estimate $\mathcal{N}_{\rm src}(z)$ we usethe Combined EIS-NVSS Survey Of Radio Sources (CENSORS;\citealt{Brookes:2008}), which forms a complete sample of radiosources brighter than 7.2\,mJy at 1.4\,GHz with spectroscopicredshifts out to cosmological distances. In \autoref{figure:zdist} weshow the distribution of CENSORS sources brighter than 10\,mJy beyonda given redshift $z$, and the corresponding analytical functionderived from the model fit of \citet{deZotti:2010}, given by\begin{equation} \mathcal{N}_{\rm src}(z) \approx 1.29 + 32.37z - 32.89z^{2} + 11.13z^{3} - 1.25z^{4},\end{equation}which we use in our analysis. For the redshifts spanned by oursimulated ASKAP survey, the fraction of background sources evolvesfrom 87\,per\,cent at $z = 0.4$ to 53\,per\,cent at $z = 1.0$.We assume that this redshift distribution applies to any sight-lineirrespective of the continuum flux density. However, this assumptionis only true if the source population in the target sample evolvessuch that the effect of distance is nullified by an increase inluminosity. Given this criterion, and the sensitivity of our simulatedsurvey, we limit our sample to sources with flux densities between 10and 1000\,mJy, which are dominated by the rapidly evolving populationof high-excitation radio galaxies and quasars(e.g. \citealt{Jackson:1999, Best:2012, Best:2014, Pracy:2016}) andfor which the redshift distribution is known to be almost independentof flux density (e.g. \citealt{Condon:1984, Condon:1998}). In\autoref{figure:nsources_tau}, we show the number of sources from thissub-sample as a function of opacity sensitivity [as defined by\autoref{equation:optical_depth_limit}], drawing random samples of theline FWHM and covering factor from the distributions shown in\autoref{figure:width_dist} and \autoref{figure:covfact_dist}. Thereare approximately 190\,000 sightlines with sufficient sensitivity todetect absorption of optical depth greater than $\tau_{5\sigma}\approx 1.0$ and 25\,000 sensitive to optical depths greater than$\tau_{5\sigma} \approx 0.1$. Since this distribution converges atoptical depth sensitivities greater than $\tau_{5\sigma} \approx 5$,the population of sources fainter than 10\,mJy, which are excludedfrom our simulated ASKAP survey, would not significantly contribute tofurther detections of absorption. Similarly, while sources brighterthan 1\,Jy are good probes of low-column-density \mbox{H\,{\sc i}}gas, they do not constitute a sufficiently large enough population tosignificantly affect the total number of absorber detections expectedin the survey and can also be safely excluded.Based on these assumptions, we can estimate the number of absorbers wewould expect to detect in our survey with ASKAP as a function of spintemperature. In \autoref{figure:ndetections_nhi}, we show the expecteddetection yield as a cumulative function of column density. We showresults for two scenarios where the spin temperature is fixed at asingle value of either 100 or 1000\,K, and the line FWHM and coveringfactors are drawn from the random distributions shown in\autoref{figure:width_dist} and \autoref{figure:covfact_dist}. We findthat for both these cases the expected number of detections is notsensitive to column densities below the DLA definition of $N_{\rm HI}= 2\times 10^{20}$\,cm$^{-2}$. We also show in\autoref{figure:ndetections_total} the expected total detection yield(integrated over all \mbox{H\,{\sc i}} column densities) as a functionof a single spin temperature $T_\mathrm{spin}$ and line FWHM$\Delta{v}_\mathrm{50}$. We find that for typical spin temperatures ofa few hundred kelvin (consistent with the typical fraction of CNMobserved in the local Universe) and a line FWHM of approximately$20$\,km\,s$^{-1}$, a wide-field 21\,cm survey with ASKAP is expectedto yield $\sim 1000$ detections. However, even moderate evolution to ahigher spin temperature in the DLA population should see significantreduction in the detection yield from this survey.\section{Inferring the average spin temperature}\label{section:spin_temp}\begin{figure}\centering\includegraphics[width=0.475\textwidth]{ndetections_nhi.pdf}\caption{The expected number of absorber detections (as a cumulative function of column density) in our simulated ASKAP survey. We show two scenarios for a single spin temperature $T_{\rm spin} = 100$ and $1000$\,K, where we have drawn random samples for the line FWHM and covering factor from the distributions shown in \autoref{figure:width_dist} and \autoref{figure:covfact_dist}. In both cases we find that the expected number of detections is not sensitive to column densities below $N_{\rm HI} = 2 \times 10^{20}$\,cm$^{-2}$, indicating that such a survey will only be sensitive to DLA systems.}\label{figure:ndetections_nhi}\end{figure}We cannot directly measure the spin temperatures of individual systemswithout additional data from either 21\,cm emission or Lyman-$\alpha$absorption. However, from \autoref{figure:ndetections_total} it isevident that the total number of absorbing systems expected to bedetected with a reasonably large 21\,cm survey is strongly dependenton the assumed value for the spin temperature. Therefore, by comparingthe actual survey yield with that expected from the known\mbox{H\,{\sc i}} distribution, we can infer the average spintemperature of the atomic gas within the DLA population for a givenredshift interval.Assuming that the total number of detections follows a Poissondistribution, the probability of detecting $\mathcal{N}$ interveningabsorbing systems is given by\begin{equation} p(\mathcal{N}|\overline{\mu}) = {\overline{\mu}^{\mathcal{N}} \over \mathcal{N}!} \mathrm{e}^{-\overline{\mu}},\end{equation}where $\overline{\mu}$ is the expected total number of detectionsgiven by the integral\begin{equation} \overline{\mu} = \iiint{\mu(T_{\rm spin},\Delta{v}_{50},c_{\rm f})\rho(T_{\rm spin},\Delta{v}_{50},c_{\rm f})\mathrm{d}T_{\rm spin}\mathrm{d}\Delta{v}_{50}\mathrm{d}c_{\rm f}},\end{equation}and $\rho$ is the distribution of systems as a function of spintemperature, line FWHM and covering factor. We assume that all threeof these variables are independent\footnote{In the case where thermal broadening contributes significantly to the velocity dispersion, and the spin temperature is dominated by collisional excitation, the assumption that these are independent may no longer hold. However, given that collisional excitation dominates in the CNM, where $T_{\rm spin} \sim 100\,\mathrm{K}$, the velocity dispersion would have to satisfy $\Delta{v}_{50} \ll 10$\,km\,s$^{-1}$ (c.f. the distribution shown in \autoref{figure:width_dist}).} so that $\rho$factorizes into functions of each. We then marginalise over thecovering factor and line width distributions shown in\autoref{figure:width_dist} and \autoref{figure:covfact_dist} so thatthe expression for $\overline{\mu}$ reduces to\begin{equation}\label{equation:harmonic_spin_temp} \overline{\mu} = \int{\mu(T_{\rm spin})\rho(T_{\rm spin})\mathrm{d}T_{\rm spin}} = \mu(\overline{T}_{\rm spin}),\end{equation}where $\overline{T}_{\rm spin}$ is the harmonic mean of the unknownspin temperature distribution, weighted by column density. This isanalogous to the spin temperature inferred from the detection ofabsorption in a single intervening galaxy averaged over severalgaseous components at different temperatures(e.g. \citealt{Carilli:1996}).\begin{figure}\centering\includegraphics[width=0.475\textwidth]{ndetections_total.pdf}\caption{The expected total number of detections in our simulated ASKAP survey, as a function of a single spin temperature ($T_{\rm spin}$) and line FWHM ($\Delta{v}_{50}$). The vertical dotted lines enclose the velocity resolution across the observed frequency band. We draw random samples for the covering factor from a uniform distribution between 0 and 1 (as shown in \autoref{figure:covfact_dist}). The contours are truncated at $\mu < 1$ for clarity.}\label{figure:ndetections_total}\end{figure}In the event of the survey yielding $\mathcal{N}$ detections, we cancalculate the posterior probability density of $\overline{T}_{\rm spin}$ using the following relationship between conditionalprobabilities\begin{equation} p(\overline{T}_{\rm spin}|\mathcal{N}) = {p(\mathcal{N}|\overline{T}_{\rm spin})p(\overline{T}_{\rm spin})\over p(\mathcal{N})},\end{equation}where $p(\overline{T}_{\rm spin})$ is our prior probability densityfor $\overline{T}_{\rm spin}$ and $p(\mathcal{N})$ is the marginalprobability of the number of detections, which can be treated as anormalizing constant. The minimally informative Jeffreys prior for themean value $\mu$ of a Poisson distribution is $1/\sqrt{\mu}$(\citealt{Jeffreys:1946})\footnote{A suitable alternative choice for the prior is the standard scale-invariant form $1/\mu$ (e.g. \citealt{Jeffreys:1961, Novick:1965, Villegas:1977}). While we find that our choice of non-informative prior has negligible effect on the spin temperature posterior for the full \mbox{H\,{\sc i}} absorption survey, as one would expect this choice becomes more important for smaller surveys. For the early-science 1000\,deg$^{2}$ survey discussed in \autoref{section:tspin_results} we find that the difference in these two priors produces a $\sim 2$ to 20\,per\,cent effect in the posterior. However, in all cases considered this change is smaller than the 68.3\,per\,cent credible interval spanned by the posterior.}. From \autoref{equation:harmonic_spin_temp} ittherefore follows that a suitable form for the non-informative spintemperature prior is $p(\overline{T}_{\rm spin}) =1/\sqrt{\overline{\mu}}$, so that\begin{equation}\label{equation:tspin_prob} p(\overline{T}_{\rm spin}|\mathcal{N}) = C^{-1}\,{\overline{\mu}^{(\mathcal{N}-1/2)} \over \mathcal{N}!} \mathrm{e}^{-\overline{\mu}}, \end{equation}where the distribution is normalised to unit total probability byevaluating the integral\begin{equation} C = \int{{\overline{\mu}^{(\mathcal{N}-1/2)} \over \mathcal{N}!} \mathrm{e}^{-\overline{\mu}}}\,\mathrm{d}\overline{T}_{\rm spin}.\end{equation}The probabilistic relationship given by \autoref{equation:tspin_prob}and the expected detection yield derived in\autoref{section:all_sky_survey} can be used as a frame-work forinferring the harmonic-mean spin temperature using the results of anyhom*ogeneous 21-cm survey. We have assumed that we can accuratelydistinguish intervening absorbing systems from those associated withthe host galaxy of the radio source. However, any 21-cm survey will beaccompanied by follow-up observations, at optical and sub-mmwavelengths, which will aid identification. Furthermore, futureimplementation of probabilistic techniques to either use photometricredshift information or distinguish between line profiles shouldprovide further disambiguation.Of course we have not yet accounted for any error in our estimate of$\overline{\mu}$, which will increase our uncertainty in$\overline{T}_\mathrm{spin}$. In the following section we discussthese possible sources of error and their effect on the result.\section{Sources of error}\label{section:errors}Our estimate of the expected number of 21\,cm absorbers is dependentupon several distributions describing the properties of the foregroundabsorbing gas and the background source distribution. For futurelarge-scale 21\,cm surveys, the accuracy to which we can infer theharmonic mean of the spin temperature distribution will eventually belimited by the accuracy to which we can measure these otherdistributions. In this section, we describe these errors and theirpropagation through to the estimate of $\overline{T}_\mathrm{spin}$,summarizing our results in \autoref{table:tspin_uncertainties}.\subsection{The covering factor}\label{section:covering_factor}\subsubsection{Deviation from a uniform distribution between 0 and 1}The fraction $c_{\rm f}$ by which the foreground gas subtends thebackground radiation source is difficult to measure directly and isthereby a significant source of error for 21\,cm absorptionsurveys. In this work, we have assumed a uniform distribution for$c_{\rm f}$, taking random values between 0 and 1. In\autoref{section:expected_number}, we tested this assumption bycomparing it with the distribution of flux density core fractions in asample of 37 quasars, used by \cite{Kanekar:2014a} as a proxy for thecovering factor. By carrying out a two-tailed KS test, we found someevidence (at the 0.05 level) that this quasar sample was inconsistentwith our assumption of a uniform distribution between 0 and1. Noticeably there seems to be an under-representation of quasars inthe Kanekar et al. sample with estimated $c_{\rm f} \lesssim 0.2$. Inthe low optical depth limit, the detection rate is dependent on theratio of spin temperature to covering factor, in which case afractional deviation in $c_{\rm f}$ will propagate as an equalfractional deviation in $\overline{T}_\mathrm{spin}$. Based on thedifference seen in the covering factor distribution of the Kanekar etal. sample and the uniform distribution, we assume that the spintemperature can deviate by as much as $\pm$10\,per\,cent.\subsubsection{Evolution with redshift}We also consider that the covering factor distribution may evolve withredshift, which would mimic a perceived evolution in the average spintemperature. Such an effect was proposed by \cite{Curran:2006b} and\cite{Curran:2012b}, who claimed that the relative change inangular-scale behaviour of absorbers and radio sources between low-and high-redshift samples could explain the apparent evolution of thespin temperature found by \cite{Kanekar:2003b}. To test for thiseffect in their larger DLA sample, \cite{Kanekar:2014a} considered asub-sample at redshifts greater than $z = 1$, for which the relativeevolution of the absorber and source angular sizes should beminimal. While the significance of their result was reduced byremoving the lower redshift absorbers from their sample, they stillfound a difference at $3.5\,\sigma$ significance between spintemperature distributions in the two DLA sub-samples separated by amedian redshift of $z = 2.683$.Future surveys with ASKAP and the other SKA pathfinders will searchfor \mbox{H\,{\sc i}} absorption at intermediate redshifts ($z \sim1$), where the relative evolution of the absorber and source angularsizes is expected to be more significant than for the higher redshiftDLA sample considered by \cite{Kanekar:2014a}. We therefore considerthe potential effect of this cosmological evolution on the inferredvalue of $\overline{T}_\mathrm{spin}$. We approximate the coveringfactor using the following model of \cite{Curran:2006b}\begin{equation}\label{equation:covering_factor} c_{f} \approx\begin{cases} \left({\theta_{\rm abs}\over \theta_{\rm src}}\right)^{2}, & \text{if}\ \theta_{\rm abs} < \theta_{\rm src}, \\ 1, & \text{otherwise},\end{cases}\end{equation}where $\theta_{\rm abs}$ and $\theta_{\rm src}$ are the angular sizesof the absorber and background source, respectively. Under thesmall-angle approximation $\theta_{\rm abs} \approx {d_{\rm abs}/D_{\rm abs}}$ and $\theta_{\rm abs} \approx {d_{\rm src}/D_{\rm src}}$, where $d_{\rm abs}$ and $D_{\rm abs}$ are thelinear size and angular diameter distance of the absorber, andlikewise $d_{\rm src}$ and $D_{\rm src}$ are the linear size andangular diameter distance of the background source. Assuming that theratio $d_{\rm abs}/d_{\rm src}$ is randomly distributed andindependent of redshift, any evolution in the covering factor istherefore dominated by relative changes in the angular diameterdistances. We calculate the expected angular diameter distance ratioat a redshift $z$ by\begin{equation} \left\langle{D_{\rm abs}\over D_{\rm src}}\right\rangle_{z} = D_{\rm abs}(z){\int_{z}^{\infty} \mathcal{N}_{\rm src}(z^{\prime})D_{\rm src}(z^{\prime})^{-1}\mathrm{d}z^{\prime}\over\int_{z}^{\infty}\mathcal{N}_{\rm src} (z^{\prime})\mathrm{d}z^{\prime}},\end{equation}which, for the source redshift distribution model given by\cite{deZotti:2010}, evolves from 0.7 at $z = 0.4$ to 1.0 at $z = 1.0$(see \autoref{figure:dang_ratio}). We note that this is consistentwith the behaviour measured by \cite{Curran:2012b} for the totalsample of DLAs observed at 21\,cm wavelengths. By applying this as acorrection to the otherwise uniformly distributed covering factor(using \autoref{equation:covering_factor}), we find that the inferredvalue of $\overline{T}_{\rm spin}$ systematically increases byapproximately 30 per\,cent.\begin{figure}\centering\includegraphics[width=0.475\textwidth]{dang_ratio.pdf}\caption{The expected redshift behaviour of $D_{\rm abs}/D_{\rm src}$ based on the \citet{deZotti:2010} model for the radio source redshift distribution.}\label{figure:dang_ratio}\end{figure}\subsection{The $\bmath{N_{\rm HI}}$ frequency distribution}\subsubsection{Uncertainty in the measurement of $f(N_{\rm HI},X)$}We assume that $f(N_{\rm HI}, X)$ is relatively well understood as afunction of redshift by interpolating between model gamma functionsfitted to the distributions at $z = 0$ and $3$. However, thesedistributions were measured from finite samples of galaxies, which ofcourse have associated uncertainties that need to be considered. Inthe case of the data presented by \cite{Zwaan:2005} and\cite{Noterdaeme:2009}, both have typical measurement uncertainties in$f(N_{\rm HI}, X)$ of approximately 10\,per\,cent over the range ofcolumn densities for which our simulated ASKAP survey is sensitive(see \autoref{figure:ndetections_nhi}). This will propagate as a10\,per\,cent fractional error in the expected number of absorberdetections, and contribute a similar percentage uncertainty in theinferred average spin temperature.\subsubsection{Correcting for 21\,cm self-absorption}In the local Universe, \cite{Braun:2012} showed that self-absorptionfrom opaque \mbox{H\,{\sc i}} clouds identified in high-resolutionimages of the Local Group galaxies M31, M33 and the Large MagellanicCloud may necessitate a correction to the local atomic mass density ofup to 30\,per\,cent. Although it is not yet clear whether this smallsample of Local Group galaxies is representative of the low-redshiftpopulation, it is useful to understand how this effect might propagatethrough to our average spin temperature measurement. We thereforereplace the gamma-function parametrization of the local $f(N_{\rm HI})$ given by \cite{Zwaan:2005} with the non-parametric valuesgiven in table\,2 of \cite{Braun:2012}, and recalculate$\overline{T}_{\rm spin}$. For an all-sky survey with the full36-antenna ASKAP we find that $\overline{T}_{\rm spin}$ increases by$\sim$30 for 100 detections and $\sim$10\,per\,cent for 1000detections. Note that the correction increases for low numbers ofdetections, which are dominated by the highest column density systems.\subsubsection{Dust obscuration bias in optically-selected DLAs}At higher redshifts, it is possible that the number density ofoptically-selected DLAs could be significantly underestimated as aresult of dust obscuration of the background quasar(\citealt{Ostriker:1984}). This would cause a reduction in the$f(\mbox{H\,{\sc i}}, X)$ measured from optical surveys, therebysignificantly underestimating the expected number of intervening21\,cm absorbers at high redshifts. The issue is further compounded bythe expectation that the highest column density DLAs ($N_{\rm HI}\gtrsim 10^{21}$\,cm$^{-2}$), for which future wide-field 21\,cmsurveys are most sensitive (see \autoref{figure:ndetections_nhi}), maycontain more dust than their less-dense counterparts.This conclusion was supported by early analyses of the existing quasarsurveys at that time (e.g. \citealt{Fall:1993}), which indicated thatup to 70\,per\,cent of quasars could be missing from optical surveysthrough the effect of dust obscuration, albeit with largeuncertainties. However, subsequent optical and infrared observationsof radio-selected quasars (e.g. \citealt{Ellison:2001};\citealt*{Ellison:2005}; \citealt{Jorgenson:2006}), which are free ofthe potential selection biases associated with these optical surveys,found that the severity of this issue was substantially over-estimatedand that there was minimal evidence in support of a correlationbetween the presence of DLAs and dust reddening. Furthermore, the\mbox{H\,{\sc i}} column density frequency distribution measured by\cite{Jorgenson:2006} was found to be consistent with theoptically-determined gamma-function parametrization of\cite{Prochaska:2005}, with no evidence of DLA systems missing fromthe SDSS sample at a sensitivity of $N_{\rm HI} \lesssim 5 \times10^{21}$\,cm$^{-2}$. Although radio-selected surveys of quasars arefree of the selection biases associated with optical surveys, they dotypically suffer from smaller sample sizes and are therefore lesssensitive to the rarer DLAs with the highest column densities.Another approach is to directly test whether optically-selectedquasars with intervening DLAs, selected from the SDSS sample, aresystematically more dust reddened than a control sample of non-DLAquasars. Comparisons in the literature are based on several differentcolour indicators, which include the spectral index(e.g. \citealt{Murphy:2004,Murphy:2016}), spectral stacking(e.g. \citealt{Frank:2010, Khare:2012}) and direct photometry(e.g. \citealt*{Vladilo:2008}; \citealt{f*ckugita:2015}). The currentstatus of these efforts is summarized by \citet{Murphy:2016}, showingbroad support for a missing DLA population at the level of$\sim$5\,per\,cent but highlighting that tension still exists betweendifferent dust measurements. No substantial evidence has yet beenfound to support a correlation between the dust reddening and\mbox{H\,{\sc i}} column density in these optically selected DLAsurveys (e.g. \citealt{Vladilo:2008, Khare:2012, Murphy:2016}).In an attempt to reconcile the differences and myriad biasesassociated with these techniques, \cite{Pontzen:2009} carried out astatistically-robust meta-analysis of the available optical and radiodata, using a Bayesian parameter estimation approach to model the dustas a function of column density and metallicity. They found that theexpected fraction of DLAs missing from optical surveys is7\,per\,cent, with fewer than 28\,per\,cent missing at 3\,$\sigma$confidence. Based on this body of work we therefore assume thatapproximately 10\,per\,cent of DLAs are missing from the SDSS sampleof \cite{Noterdaeme:2009} and consider the affect on our estimate of$\overline{T}_{\rm spin}$. We further assume that there is nodependance on column density, an assumption which is supported by theaforementioned observational data for the range of column densities towhich our 21\,cm survey is sensitive. We find that increasing thehigh-redshift column density frequency distribution by 10\,per\,centintroduces a systematic increase of approximately 3\,per\,cent in theexpected number of detections for the redshifts covered by our ASKAPsurveys. We note that this error will increase significantly for21\,cm surveys at higher redshifts where the optically derived$f(\mbox{H\,{\sc i}}, X)$ dominates the calculation of the expecteddetection rate.\subsection{The radio source background}As described in \autoref{section:all_sky_survey}, we weight thecomoving path-length for each sight-line by a statistical redshiftdistribution in order to account for evolution in the radio sourcebackground. We use the parametric model of \cite{deZotti:2010}, whichis derived from fitting the measured redshifts of \cite{Brookes:2008}for CENSORS sources brighter than 10\,mJy, and assume that thisapplies to all sources in the range 10 - 1000\,mJy. In\autoref{figure:zdist}, we show the cumulative distribution of sourceslocated behind a given redshift and the associated measurementuncertainty given by the errorbars. For the intermediate redshiftscovered by the ASKAP survey, the fractional uncertainty in thisdistribution increases from $\sigma_{\mathcal{F}_{\rm src}}/\mathcal{F}_{\rm src} \approx 3.5$ to 8\,per\,cent between$z = 0.4$ and 1.0, which propagates through to a similar fractionaluncertainty in $\overline{T}_{\rm spin}$. However, for higherredshifts this fractional uncertainty increases rapidly at $z > 2$, tomore than 50\,per\,cent at $z = 3$, reflecting the paucity of opticalspectroscopic data for the high-redshift radio sourcepopulation. Understanding how the radio source population isdistributed at lower flux densities and at higher redshifts istherefore a concern for the future 21\,cm absorption surveysundertaken with the SKA mid- and low-frequency telescopes (see\citealt{Kanekar:2004} and \citealt*{Morganti:2015} for reviews).\begin{table} \begin{threeparttable} \caption{An account of errors in our estimate of $\overline{T}_{\rm spin}$ due to the accuracy to which we can determine the expected number of absorber detections.}\label{table:tspin_uncertainties} \begin{tabular}{l@{\hspace{0.05in}}l@{\hspace{0.05in}}l@{\hspace{0.05in}}l@{\hspace{0.05in}}l} \hline & Source of error & $\mathrm{err}(\overline{T}_{\rm spin})$ & Refs. \\ & & [per\,cent] & \\ \hline Covering factor & Distribution uncertainty & $\pm10$ & $a$ \\ Covering factor & Systematic evolution & +30 & $a$, $b$\\ $f(N_{\rm HI}, X)$ & Measurement uncertainty & $\pm10$ & $c, d$\\ Low-$z$ $f(N_{\rm HI}, X)$ & Systematic self-absorption & $+(10-30)$ & $e$ \\ High-$z$ $f(N_{\rm HI}, X)$ & Systematic dust-obscuration & $+3$ & $f$, $g$ \\ $\mathcal{F}_{\rm src}(z^{\prime} \geq z)$ & Measurement uncertainty & $\pm 5$ & $h$, $i$ \\ \hline \end{tabular} \begin{tablenotes} \item[] References: $^{a}${\citet{Kanekar:2014a}}, $^{b}${\citet{Curran:2012b}}, $^{c}${\citet{Zwaan:2005}}, $^{d}${\citet{Noterdaeme:2009}} , $^{e}${\citet{Braun:2012}}, $^{f}${\citet{Pontzen:2009}}, $^{g}${\citet{Murphy:2016}}, $^{h}${\citet{Brookes:2008}}, $^{i}${\citet{deZotti:2010}}. \end{tablenotes}\end{threeparttable}\end{table}\section{Expected results for future 21-cm absorption surveys}\label{section:tspin_results}\begin{figure}\centering\includegraphics[width=0.475\textwidth]{tspin_prob.pdf}\caption{The posterior probability density of the average spin temperature, as a function of absorber detection yield ($\mathcal{N}$). We show results for our simulated all-southern-sky survey with 2-h per pointing using the full 36-antenna ASKAP (top panel) and a smaller 1000\,deg$^{2}$ survey with 12-h per pointing and 12 antennas of ASKAP (bottom panel). The dashed curves show the cumulative effect of the systematic errors discussed in \autoref{section:errors}. $\overline{\mathcal{F}}_{\rm CNM}$ is the average CNM fraction assuming a simple two-phase neutral ISM with $T_{\rm spin,CNM} = 100$\,K and $T_{\rm spin,WNM} = 1800$\,K (\citealt{Liszt:2001}).}\label{figure:tspin_prob}\end{figure}In the top panel of \autoref{figure:tspin_prob} we show the results ofapplying our method for inferring $\overline{T}_{\rm spin}$ to thesimulated all-southern-sky \mbox{H\,{\sc i}} absorption survey withASKAP described in \autoref{section:all_sky_survey}. We account forthe uncertainties in the expected detection rate $\overline{\mu}$,discussed in \autoref{section:errors}, by using a Monte Carlo approachand marginalizing over many realizations. A yield of 1000 absorbersfrom such a survey would imply an average spin temperature of$\overline{T}_\mathrm{spin} =127^{+14}_{-14}\,(193^{+23}_{-23})$\,K\footnote{We give the 68.3\,per\,cent interval about the median value measured from the posterior distributions shown in \autoref{figure:tspin_prob}.},where values in parentheses denote the alternative posteriorprobability resulting from the systematic errors discussed in\autoref{section:errors}. This scenario would indicate that a largefraction of the atomic gas in DLAs at these intermediate redshifts isin the classical stable CNM phase. Conversely, a yield of only 100detections would imply that $\overline{T}_\mathrm{spin} =679^{+64}_{-65}\,(1184^{+116}_{-120})$\,K, indicating that less than10\,per\,cent of the atomic gas is in the CNM and that the bulk of theneutral gas in galaxies is significantly different at intermediateredshifts compared with the local Universe.We also consider the effect of reducing the sky area and array size,which is relevant for planned early science surveys with ASKAP andother SKA pathfinder telescopes. In the bottom panel of\autoref{figure:tspin_prob}, we show the spin temperatures inferredwhen observing a random 1000\,deg$^{2}$ field for 12\,h per pointing,between $z_{\rm HI} = 0.4$ and $1.0$, using a 12-antenna version ofASKAP. We find that detection yields of 30 and 3 from such a surveywould give inferred spin temperatures of $\overline{T}_{\rm spin}=134^{+23}_{-27}\,(209^{+40}_{-47})$ and$848^{+270}_{-430}\,(1535^{+513}_{-837})$\,K, respectively. Thesignificant reduction in telescope sensitivity and sky-area,compensated by the increase in integration time per pointing plannedfor early-science, results in a factor of 30 decrease in the expectednumber of detections and therefore an increase in the sample varianceand uncertainty in $\overline{T}_{\rm spin}$. However, this resultdemonstrates that we expect to be able to distinguish between thelimiting cases of CNM-rich or deficient DLA populations even duringthe early-science phases of the SKA pathfinders. For example 30detections with the early ASKAP survey rules out an average spintemperature of 1000\,K at high probability.\section{Conclusions}We have demonstrated a statistical method for measuring the averagespin temperature of the neutral ISM in distant galaxies, using theexpected detection yields from future wide-field 21\,cm absorptionsurveys. The spin temperature is a crucial property of the ISM thatcan be used to determine the fraction of the cold ($T_{\rm k} \sim100$\,K) and dense ($n \sim 100$\,cm$^{-2}$) atomic gas that providessites for the future formation of cold molecular gas clouds and starformation. Recent 21\,cm surveys for \mbox{H\,{\sc i}} absorption in\mbox{Mg\,{\sc ii}} absorbers and DLAs towards distant quasars haveyielded some evidence of an evolution in the average spin temperaturethat might reveal a decrease in the fraction of cold dense atomic gasat high redshift (e.g. \citealt{Gupta:2009, Kanekar:2014a}).By combining recent specifications for ASKAP, with availableinformation for the population of background radio sources, we showthat strong statistical constraints (approximately $\pm10$\,per\,cent)in the average spin temperature can be achieved by carrying out ashallow 2-h per pointing survey of the southern sky between redshiftsof $z = 0.4$ and $1.0$. However, we find that the accuracy to which wecan measure the average spin temperature is ultimately limited by theaccuracy to which we can measure the distribution of the coveringfactor, the $N_{\rm HI}$ frequency distribution function and theevolution of the radio source population as a function of redshift. Byimproving our understanding of these distributions we will be able toleverage the order-of-magnitude increases in sensitivity and redshiftcoverage of the future SKA telescope, allowing us to measure theevolution of the average spin temperature to much higher redshifts.\section*{Acknowledgements} We thank Robert Allison, Elaine Sadler and Michael Pracy for usefuldiscussions, and the anonymous referee for providing comments thathelped improve this paper. JRA acknowledges support from a BoltonFellowship. We have made use of \texttt{Astropy}, acommunity-developed core \texttt{Python} package for astronomy(\citealt{Astropy:2013}); NASA's Astrophysics Data SystemBibliographic Services; and the VizieR catalogue access tool, operatedat CDS, Strasbourg, France.\bibliographystyle{mnras}

{'timestamp': '2016-08-09T02:04:29', 'yymm': '1607', 'arxiv_id': '1607.04828', 'language': 'en', 'url': 'https://arxiv.org/abs/1607.04828'}

arxiv

\section{Introduction}Given $\rho>0$, we consider the problem\begin{equation}\label{eq:main_prob_U}\begin{cases}-\Delta U + \lambda U = |U|^{p-1}U & \text{in }\Omega,\smallskip\\\int_\Omega U^2\,dx = \rho, \quad U=0 & \text{on }\partial\Omega,\end{cases}\end{equation}where $\Omega\subset{\mathbb{R}}^N$ is a Lipschitz, bounded domain, $1<p<2^*-1$, $\rho>0$ is a fixedparameter, and both $U\in H^1_0(\Omega)$ and $\lambda\in{\mathbb{R}}$ are unknown. More precisely,we investigate conditions on $p$ and $\rho$ (and also $\Omega$) for the solvability ofthe problem.The main interest in \eqref{eq:main_prob_U} relies on the investigation of standing wavesolutions for the nonlinear Schr\"odinger equation\[i\frac{\partial \Phi}{\partial t}+\Delta \Phi+ |\Phi|^{p-1}\Phi=0,\qquad(t,x)\in {\mathbb{R}}\times \Omega\]with Dirichlet boundary conditions on $\partial\Omega$. This equation appears in severaldifferent physical models, both in the case $\Omega={\mathbb{R}}^N$ \cite{MR2002047}, and on boundeddomains \cite{MR1837207}. In particular, the latter case appears in nonlinear optics and inthe theory of Bose-Einstein condensation, also as a limiting case of the equation on ${\mathbb{R}}^N$with confining potential. When searching for solutions having thewave function $\Phi$ factorized as $\Phi(x,t)=e^{i\lambda t} U(x)$, one obtainsthat the real valued function $U$ must solve\begin{equation}\label{eq:NLS}-\Delta U + \lambda U = |U|^{p-1}U ,\qquad U\in H^1_0(\Omega),\end{equation}and two points of view are available. The first possibility is to assign the chemicalpotential $\lambda\in{\mathbb{R}}$, and search for solutions of \eqref{eq:NLS} as critical points of therelated action functional. The literature concerning this approach is huge and we do not evenmake an attempt to summarize it here. On the contrary, we focus on the second possibility,which consists in considering $\lambda$ as part of the unknown and prescribing the mass (orcharge) $\|U\|_{L^2(\Omega)}^2$ as a natural additional condition. Up to our knowledge,the only previous paper dealing with this case, in bounded domains, is \cite{MR3318740},which we describe below. The problem of searching for normalized solutions in ${\mathbb{R}}^N$,with non-hom*ogeneous nonlinearities, is more investigated \cite{MR3009665,MR1430506},even though the methods used there can not be easily extended to bounded domains, wheredilations are not allowed. Very recently, also the case of partial confinement has beenconsidered \cite{BeBoJeVi_2016}.Solutions of \eqref{eq:main_prob_U} can be identified with critical points of theassociated energy functional\[\mathcal{E}(U) = \frac12\int_\Omega|\nabla U|^2\,dx - \frac{1}{p+1} \int_\Omega|U|^{p+1}\,dx\]restricted to the mass constraint\[{\mathcal{M}}_\rho=\{U\in H_0^1(\Omega) : \|U\|_{L^2(\Omega)}=\rho\},\]with $\lambda$ playing the role of a Lagrange multiplier.A cricial role in the discussion of the above problem is played by the Gagliardo-Nirenberginequality: for any $\Omega$ and for any $v\in H^1_0(\Omega)$,\begin{equation}\label{sobest}\|v\|^{p+1}_{L^{p+1}(\Omega)} \leq C_{N,p} \| \nabla v \|_{L^2(\Omega)}^{N(p-1)/2}\| v \|_{L^2(\Omega)} ^{(p+1)-N(p-1)/2},\end{equation}the equality holding only when $\Omega={\mathbb{R}}^N$ and $v=Z_{N,p}$, the positive solution of $-\Delta Z + Z = Z^{p}$ (which is unique up totranslations \cite{MR969899}).Accordingly, the exponent $p$ can be classified in relation with the so called\emph{$L^2$-critical exponent} $1+4/N$ (throughout all the paper, $p$ will be alwaysSobolev-subcritical and its criticality will be understood in the $L^2$ sense).Indeed we have that ${\mathcal{E}}$ is bounded below and coercive on ${\mathcal{M}}_\rho$ if and only ifeither $p$ is subcritical, or it is critical and $\rho$ is sufficiently small.The recent paper \cite{MR3318740} deals with problem \eqref{eq:main_prob_U} in the case of the spherical domain $\Omega = B_1$, when searching for positive solutions $U$.In particular, it is shown that the solvability of \eqref{eq:main_prob_U} is strongly influenced by the exponent $p$, indeed:\begin{itemize} \item in the subcritical case $1<p<1+4/N$, \eqref{eq:main_prob_U} admits a unique positive solution for every $\rho>0$; \item if $p=1+4/N$ then \eqref{eq:main_prob_U} admits a unique positive solution for \[ 0<\rho<\rho^*=\left(\frac{p+1}{2C_{N,p}}\right)^{N/2}=\|Z_{N,p}\|^2_{L^2({\mathbb{R}}^N)}, \] and no positive solutions for $\rho\geq\rho^*$; \item finally, in the supercritical regime $1+4/N<p<2^*-1$, \eqref{eq:main_prob_U} admits positive solutions if and only if $0<\rho\leq\rho^*$ (the threshold $\rho^*$ depending on $p$), and such solutions are at least two for $\rho<\rho^*$.\end{itemize}In this paper we carry on such analysis, dealing with a general domain $\Omega$ and with solutions whichare not necessarily positive.More precisely, let us recall that for any $U$ solving \eqref{eq:main_prob_U} for some $\lambda$, it is well-defined the Morse index\[m(U) = \max\left\{k : \begin{array}{l}\exists V\subset H^1_0(\Omega),\,\dim(V)= k:\forall v\in V\setminus\{0\}\smallskip\\\displaystyle\int_\Omega |\nabla v|^2 + \lambda v^2 - p|U|^{p-1}v^2\,dx<0\end{array}\right\}\in{\mathbb{N}}.\]Then, if $\Omega=B_1$, it is well known that a solution $U$ of \eqref{eq:main_prob_U} ispositive if and only if $m(U)=1$. Under this perspective, the results in \cite{MR3318740}can be read in terms of Morse index one--solutions, rather than positive ones: introducing the sets of admissible masses\[{\mathfrak{A}}_k ={\mathfrak{A}}_k(p,\Omega) := \left\{\rho>0 : \begin{array}{l} \eqref{eq:main_prob_U}\text{ admits a solution $U$ (for some $\lambda$)}\\ \text{having Morse index }m(U)\leq k\end{array} \right\},\]then \cite{MR3318740} implies that ${\mathfrak{A}}_1(p,B_1)$ is a bounded interval ifand only if $p$ is critical or supercritical, while ${\mathfrak{A}}_1(p,B_1)={\mathbb{R}}^+$ in the subcritical case.On the contrary, when considering general domains and higher Morse index, the situation may becomemuch more complicated. We collect some examples in the following remark.\begin{remark}\label{rem:specialdomains}In the case of a symmetric domain, one can use any solution as a building block to construct other solutions witha more complex behavior, obtaining the so-called necklace solitary waves. Such kind of solutions are constructedin \cite{MR3426917}, even though in such paper the focus is on stability, rather than on normalization conditions.For instance, by scaling argument, any Dirichlet solution of $-\Delta U + \lambda U = |U|^{p-1}U$ in a rectangle $R=\prod_{i=1}^N(a_i,b_i)$ canbe scaled to a solution of $-\Delta U + k^2\lambda U = |U|^{p-1}U$ in $R/k$, $k\in{\mathbb{N}}_+$, and then $k^N$ copies of it can bejuxtaposed, with alternating sign. In this way one obtains a new solution on $R$ having $k^{4/(p-1)}$ times the mass of the starting one,and eventually solutions in $R$ with arbitrarily high mass (but with higher Morse index) can be constructed even in the criticaland supercritical case. An analogous construction can be performed in the disk, using solutions in circular sectors as building blocks,even though in this case explicit bounds on the mass obtained are more delicate.Also, instead of symmetric domains, singular perturbed ones can be considered, such as dumbbell domains\cite{MR949628}: for instance, using \cite[Theorem 3.5]{MR2997381}, one can show that for any $k$, there exists a domain $\Omega$,which is close in a suitable sense to the disjoint union of $k$ domains, such that \eqref{eq:main_prob_U} has a \emph{positive}solution on $\Omega$ with Morse index $k$ and $\rho=\rho_k\to+\infty$ as $k\to+\infty$.This kind of results justifies the choice of classifying the solutions in terms of their Morse index, ratherthan in terms of their nodal properties.\end{remark}Motivated by the previous remark, the first question we address in this paper concerns the boundedness of ${\mathfrak{A}}_k$.We provide the following complete classification.\begin{theorem}\label{thm:bbd_index}For every $\Omega\subset{\mathbb{R}}^N$ bounded $C^1$ domain, $k\ge1$, $1<p<2^*-1$,\[\sup{\mathfrak{A}}_k(p,\Omega) < +\infty\qquad\iff\qquadp \ge 1+\frac{4}{N}.\]\end{theorem}The proof of such result, which is outlined in Section \ref{sec:blow-up}, is obtainedby a detailed blow-up analysis of sequences of solutions with bounded Morse index, viasuitable a priori pointwise estimates (see \cite{MR2063399}). In this respect, theregularity assumption on $\partial\Omega$ simplifies the treatment of possibleconcentration phenomena towards the boundary. The argument, which holdsfor solutions which possibly change sign, is inspired by \cite{MR2825606}, where thecase of positive solutions is treated.Once Theorem \ref{thm:bbd_index} is established, in case $p\geq 1 + 4/N$ two questions arise, namely:\begin{enumerate} \item is it possible to provide lower bounds for $\sup{\mathfrak{A}}_k$? Is it true that $\sup{\mathfrak{A}}_k$ is strictly increasing in $k$, or, at least, that $\sup{\mathfrak{A}}_k > \sup{\mathfrak{A}}_1$ for some $k$? \item is \eqref{eq:main_prob_U} solvable for every $\rho\in(0,\sup{\mathfrak{A}}_k)$, or at least can we characterize some subinterval of solvability?\end{enumerate}It is clear that both issues can be addressed by characterizingvalues of $\rho$ for which existence (and multiplicity) of solutions withbounded Morse index can be guaranteed. To this aim, it can be useful to restateproblem \eqref{eq:main_prob_U} as\begin{equation}\label{eq:main_prob_u}\begin{cases}-\Delta u + \lambda u = \mu|u|^{p-1}u & \text{in }\Omega,\\\int_\Omega u^2\,dx = 1, \quad u=0 & \text{on }\partial\Omega,\end{cases}\qquad\text{where}\quad\begin{cases}U=\sqrt{\rho} u\\\mu = \rho^{(p-1)/2},\end{cases}\end{equation}where now $\mu>0$ is prescribed. Since\begin{equation}\label{Emu}\text{both } \mathcal{E}_{\mu}(u) := \frac{1}{2}\int_{\Omega}|\nabla u|^2- \frac{\mu}{p+1}\int_{\Omega}| u|^{p+1}\qquad\text{and }{\mathcal{M}}={\mathcal{M}}_1=\{u : \|u\|_{L^2(\Omega)}=1\}\end{equation}are invariant under the ${\mathbb{Z}}_2$-action of the involution $u\mapsto -u$, solutions of\eqref{eq:main_prob_u} can be found via min-max principles in the framework of indextheories (see e.g. \cite[Ch. II.5]{St_2008}). Notice that in the supercritical case${\mathcal{E}}_\mu$ is not bounded from below on ${\mathcal{M}}$. Following \cite{MR3318740}, it can beconvenient to parameterize solutions to \eqref{eq:main_prob_u} with respect to the$H^1_0$-norm, therefore we introduce the sets\begin{equation}\label{eq:defBU}\mathcal{B}_\alpha:=\left\{u\in {\mathcal{M}}:\,\int_\Omega |\nabla u|^2\,dx<\alpha\right\},\quad\quad\mathcal{U}_\alpha:=\left\{u\in {\mathcal{M}}:\,\int_\Omega |\nabla u|^2\,dx=\alpha\right\}.\end{equation}Introducing the first Dirichlet eigenvalue of $-\Delta$ in $H^1_0(\Omega)$,$\lambda_1(\Omega)$, we have that the sets above are non-empty whenever $\alpha>\lambda_1(\Omega)$.Since we are interested in critical pointshaving Morse index bounded from above, following \cite{MR968487,MR954951,MR991264} weintroduce the following notion of genus.\begin{definition}\label{def:genus}Let $A\subset H^1_0(\Omega)$ be a closed set, symmetric with respect to the origin (i.e.$-A=A$). We define the \emph{genus} $\gamma$ of a $A$ as\[\gamma(A) := \sup\{m : \exists h \in C({\mathbb{S}}^{m-1};A),\, h(-u)=-h(u)\}.\]Furthermore, we define\[\Sigma_{\alpha}=\{A\subset \overline{\mathcal{B}}_\alpha: A\text{ is closed and }-A=A\},\qquad\Sigma^{(k)}_{\alpha}=\{A\in \Sigma_{\alpha} : \gamma(A)\ge k\},\]\end{definition}We remark that this notion of genus is different from theclassical one of \emph{Krasnoselskii genus}, which is well suited for estimatesof the Morse index from below, rather than above. Nonetheless, $\gamma$ shares withthe Krasnoselskii genus most of the main properties of an index\cite{MR0163310,MR0065910}. In particular, by the Borsuk-Ulam Theorem, any set $A$homeomorphic to the sphere ${\mathbb{S}}^{m-1} := \partial B_1 \subset {\mathbb{R}}^m$ has genus$\gamma(A) = m$. Furthermore, we show in Section \ref{sec:2const} that $\Sigma^{(k)}_{\alpha}$ is not empty,provided $\alpha>\lambda_k(\Omega)$ (the $k$-th Dirichlet eigenvalue of $-\Delta$ in $H^1_0(\Omega)$).Equipped with this notion of genus we provide two different variational principlesfor solutions of \eqref{eq:main_prob_u} (and thus of \eqref{eq:main_prob_U}). The firstone is based on a variational problem with \emph{two constraints}, which was exploited as themain tool in proving the results in \cite{MR3318740}.\begin{theorem}\label{thm:genus_2constr}Let $k\geq1$ and $\alpha>\lambda_{k}(\Omega)$. Then\begin{equation}\label{maxmin}M_{\alpha,\,k}:= \sup_{A\in\Sigma^{(k)}_{\alpha}}\inf_{u\in A}\int_{\Omega}|u|^{p+1}\end{equation}is achieved on ${\mathcal{U}}_\alpha$, and there exists a critical point$u_\alpha\in {\mathcal{M}}$ suchthat, for some $\lambda_\alpha\in{\mathbb{R}}$ and $\mu_\alpha>0$,\begin{equation}\label{lagreq}\int_\Omega|\nabla u_\alpha|^2 = \alpha\qquad\text{and}\qquad -\Delta u_\alpha+\lambda_\alpha\,u_\alpha=\mu_\alpha |u_\alpha|^{p-1}u_\alpha\quad \text{in }\Omega.\end{equation}\end{theorem}As a matter of fact, the results in \cite{MR3318740} were obtained by a detailed analysis of the map $\alpha \mapsto \mu_\alpha$ in the case $k=1$, i.e.when dealing with\[M_{\alpha,1} = \max\left\{\|u\|_{L^{p+1}}^{p+1} : \|u\|_{L^2}^2=1,\,\|\nabla u\|_{L^2}^2=\alpha \right\}.\]In the present paper we do not investigate the properties of the map$\alpha \mapsto \mu_\alpha$ for general $k$, but we rather prefer to exploit thecharacterization of $M_{\alpha,k}$ in connection witha second variational principle, which deals with only\emph{one constraint}.\begin{theorem}\label{thm:genus_1constr}Let $1+{N}/{4}\leq p<2^*-1$. There exists a sequence $(\hat \mu_k)_k$(depending on $\Omega$ and $p$) such that, for every $k\geq 1$ and $0<\mu<\hat \mu_k$, thevalue\begin{equation}\label{infsuplev}c_k:= \inf_{A\in\Sigma^{(k)}_{\alpha}}\sup_{A}{\mathcal{E}}_\mu,\end{equation}is achieved in $\mathcal{B}_\alpha$, for a suitable $\alpha>\lambda_{k}(\Omega)$. Furthermore there exists a critical point $u_\mu\in {\mathcal{M}}$ suchthat, for some $\lambda_\mu\in{\mathbb{R}}$,\[-\Delta u_\mu+\lambda_\mu\,u_\mu=\mu |u_\mu|^{p-1}u_\mu\quad \text{in }\Omega,\]$\|\nabla u\|_{L^2}^2<\alpha$, and $m(u_\mu)\le k$.\end{theorem}\begin{remark}Of course, if $p<1+4/N$, the above theorem holds with $\hat\mu_k=+\infty$ for every $k$.\end{remark}\begin{corollary}Let $\hat \rho_k := \hat \mu_k^{2/(p-1)}$. Then\[(0,\hat\rho_k) \subset {\mathfrak{A}}_k.\]\end{corollary}The link between Theorem \ref{thm:genus_2constr} and Theorem \ref{thm:genus_1constr} is that we can provide explicit estimatesof $\hat \mu_k$ (and hence of $\hat\rho_k$) in terms of the map $\alpha\mapsto M_{\alpha,k}$ (see Section \ref{sec:1const}).We stress that the aboveresults hold for any Lipschitz $\Omega$. As a first consequence, this allows to extend theexistence result in \cite{MR3318740} to non-radial domains.\begin{theorem}\label{thm:intro_GS}For every $0<\rho<\hat\rho_1=\hat\rho_1(\Omega,p)$ problem \eqref{eq:main_prob_U} admits a solution which is a local minimum ofthe energy ${\mathcal{E}}$ on ${\mathcal{M}}_\rho$. In particular, $U$ is positive, has Morse index one and the associated solitary wave is orbitallystable.Furthermore, for every Lipschitz $\Omega$,\begin{itemize}\item $\displaystyle 1<p<1+\frac{4}{N} \implies \hat\rho_1\left(\Omega,p\right)= +\infty$,\item $\displaystyle p=1+\frac{4}{N} \implies \hat\rho_1\left(\Omega,p\right)\geq \|Z_{N,p}\|^2_{L^2({\mathbb{R}}^N)}$,\item $\displaystyle 1+\frac{4}{N}<p<2^*-1 \implies \hat\rho_1\left(\Omega,p\right)\geq D_{N,p} \lambda_1(\Omega)^{\frac{2}{p-1}-\frac{N}{2}}$,\end{itemize}where the universal constant $D_{N,p}$ is explicitly written in terms of $N$ and $p$ inSection \ref{sec:1const}.\end{theorem}\begin{remark}\label{rem:introGS}Of course, in the subcritical and critical cases, $c_1$ is actually a global minimum.Furthermore, the lower bound for the supercritical case agrees with that of the critical onesince, as shown in Section \ref{sec:1const}, $D_{N,1+4/N} = \|Z_{N,p}\|^2_{L^2({\mathbb{R}}^N)}$(and $\lambda_1(\Omega)$ is raised to the $0^{\text{th}}$-power). Notice that the estimate for the supercritical case is new also in the case $\Omega=B_1$.\end{remark}We observe that the exponent of $\lambda_1(\Omega)$ in the supercritical threshold is negative,therefore such threshold decreases with the size of $\Omega$.Once the first thresholds have been estimated, we turn to the higher ones: by exploitingthe relations between $M_{\alpha,k}$ and $c_k$, we can show that the thresholds obtainedfor Morse index one--solutions in Theorem \ref{thm:intro_GS} can be increased, byconsidering higher Morse index--solutions, at least for some exponent.\begin{proposition}\label{thm:intro_3>1}For every $\Omega$ and $1<p<2^*-1$,\[\hat\rho_3\left(\Omega,p\right) \geq 2 \cdot D_{N,p} \lambda_3(\Omega)^{\frac{2}{p-1}-\frac{N}{2}}.\]\end{proposition}\begin{remark}In the critical case, the lower bound for $\hat\rho_3$ provided by Proposition \ref{thm:intro_3>1} is twice that for $\hat\rho_1$ obtained in Theorem \ref{thm:intro_GS}.By continuity, the estimate for $\hat\rho_3$ is larger than that for $\hat\rho_1$ also when$p$ is supercritical, but not too large. To quantify such assertion, we can use Yang's inequality\cite{MR1894540,MR2262780}, which implies that for every $\Omega$ it holds\[\lambda_3(\Omega)\leq \left(1+\frac{N}{4}\right)2^{2/N} \lambda_1(\Omega).\]We deduce that $2 \cdot D_{N,p} \lambda_3(\Omega)^{\frac{2}{p-1}-\frac{N}{2}}\geq D_{N,p} \lambda_1(\Omega)^{\frac{2}{p-1}-\frac{N}{2}}$ whenever\[p\leq 1+\frac{4}{N} + \frac{8}{N^2\log_2\left(1+\frac{4}{N}\right)}.\]In particular, the physically relevant case $N=3$, $p=3$ is covered. Furthermore, if$N\geq 7$, the above condition holds for every $p<2^*-1$.\end{remark}Beyond existence results for\eqref{eq:main_prob_U}, also multiplicity results can be achieved. A first generalconsideration, with this respect, is that Theorem \ref{thm:genus_1constr} holds truealso when using the standard Krasnoselskii genus instead of $\gamma$; this allows toobtain critical points having Morse index bounded from below (see\cite{MR968487,MR954951,MR991264}), and therefore to obtain infinitely many solutions,at least when $\rho$ is less than some threshold. More specifically, we can alsoprove the existence of a second solution in the supercritical case, thus extending toany $\Omega$ the multiplicity result obtained in \cite{MR3318740} for the ball.Indeed, on the onehand, in the supercritical case ${\mathcal{E}}_\mu$ is unbounded from below; on the other handthe solution obtained in Theorem \ref{thm:genus_1constr}, for $k=1$, is a local minimum.Thus the Mountain Pass Theorem \cite{MR0370183} applies on $\mathcal{M}$, and a second solution can befound for $\mu<\hat\mu_1$, see Proposition \ref{mpcritlev} for further details (and alsoRemark \ref{rem:further_crit_lev} for an analogous construction for $k\ge2$).To conclude this introduction, let us mention that the explicit lower bounds obtained inTheorem \ref{thm:intro_GS} can be easily applied in order to gainmuch more information also in the case of special domains, as those considered inRemark \ref{rem:specialdomains}. For instance, we can prove then following.\begin{theorem}\label{pro:symm}Let $\Omega=B$ be a ball in ${\mathbb{R}}^N$. Then\[p<1+\frac{4}{N-1}\quad\implies\quad\text{\eqref{eq:main_prob_U} admits a solution for every }\rho>0.\]An analogous result holds when $\Omega=R$ is a rectangle, without further restrictions on$p<2^*-1$.\end{theorem}Therefore our starting problem in $\Omega=B$ can be solved for any mass value also in the critical andsupercritical regime, at least for $p$ smaller than this further critical exponent $1+4/(N-1) > 1+ 4/N$.Of course, higher masses require higher Morse index--solutions. In particular, since by \cite{MR3318740}we know that ${\mathfrak{A}}_1(B,1+4/N) = (0,\|Z_{N,p}\|_{L^2})$, we have that forlarger masses, even though no positive solution exists, nodal solutions with higherMorse index can be obtained: in such cases \eqref{eq:main_prob_U} admits \emph{nodal groundstates with higher Morse index}.The paper is structured as follows: in Section \ref{sec:blow-up} we perform a blow-up analysis ofsolutions with bounded Morse index, in order to prove Theorem \ref{thm:bbd_index}; Section \ref{sec:2const}is devoted to the analysis of the variational problem with two constraints \eqref{maxmin}and to the proof of Theorem \ref{thm:genus_2constr};that of Theorems \ref{thm:genus_1constr}, \ref{thm:intro_GS} and Proposition\ref{thm:intro_3>1} is developedin Section \ref{sec:1const}, by means of the variational problem with one constraint\eqref{infsuplev}; finally, Section\ref{sec:symm} contains the proof of Theorem \ref{pro:symm}.\textbf{Notation.} We use the standard notation$\{\varphi_k\}_{k\geq1}$ for a basis of eigenfunctions of the Dirichlet laplacian in $\Omega$,orthogonal in $H^1_0(\Omega)$ and orthonormal in $L^2(\Omega)$. Such functions are ordered in sucha way that the corresponding eigenvalues $\lambda_k(\Omega)$ satisfy\[0<\lambda_1(\Omega)<\lambda_2(\Omega)\leq\lambda_3(\Omega)\leq\dots,\]and $\varphi_1$ is chosen to be positive on $\Omega$. $C_{N,p}$ denotes the universal constant inthe Gagliardo-Nirenberg inequality \eqref{sobest}, which is achieved (uniquely, up to translations and dilations)by the positive, radially symmetric function $Z_{N,p}\in H^1({\mathbb{R}}^N)$, with\[\|Z_{N,p}\|^2_{L^2({\mathbb{R}}^N)}=\left(\frac{p+1}{2C_{N,p}}\right)^{N/2}.\]Finally, $C$ denotes every (positive) constant we need not to specify, whose value may change also within the same formula.\section{Blow-up analysis of solutions with bounded Morse index}\label{sec:blow-up}Throughout this section we will deal with a sequence $\{(u_n,\mu_n,\lambda_n)\}_n \subset H^1_0(\Omega)\times{\mathbb{R}}^+\times{\mathbb{R}}$ satisfying\begin{equation}\label{eq:auxiliary_n}-\Delta u_n+\lambda_n u_n=\mu_n |u_n|^{p-1}u_n,\qquad\int_\Omega u_n^2\, dx=1,\qquad \int_\Omega |\nabla u_n|^2\, dx=:\alpha_n.\end{equation}To start with, we recall the following result (actually, in \cite{MR3318740}, the result is stated for positive solution, but the proofdoes not require such assumption).\begin{lemma}[{\cite[Lemma 2.5]{MR3318740}}]\label{lemma:case_alpha_n_bounded}Take a sequence $\{(u_n,\mu_n,\lambda_n)\}_n$ as in \eqref{eq:auxiliary_n}. Then\[\{\alpha_n\}_n \text{ bounded}\qquad\implies\qquad\{\lambda_n\}_n,\,\{\mu_n\}_n\text{ bounded}.\]\end{lemma}Next we turn to the study of sequences having arbitrarily large $H^1_0$-norm. In particular, we will focus on sequences of solutions having a common upperbound on the Morse index\[m(u_n) = \max\left\{k : \begin{array}{l}\exists V\subset H^1_0(\Omega),\,\dim(V)= k:\forall v\in V\setminus\{0\}\smallskip\\\displaystyle\int_\Omega |\nabla v|^2 + \lambda_n v^2 - p\mu_n|u_n|^{p-1}v^2\,dx<0\end{array}\right\}.\]Throughout this section we will assume that\begin{equation}\label{eq:mainass_secMorse}\text{the sequence }\{(u_n,\mu_n,\lambda_n)\}_n\text{ satisfies \eqref{eq:auxiliary_n},with }\alpha_n\to+\infty\text{ and }m(u_n)\leq \bar k,\end{equation}for some $\bar k\in{\mathbb{N}}$ not depending on $n$.\begin{lemma}\label{lem:lambda_bdd_below}Let \eqref{eq:mainass_secMorse} hold. Then\(\lambda_n \geq -\lambda_{\bar k}(\Omega).\)\end{lemma}\begin{proof}Assume, to the contrary, that for some $n$ it holds $\lambda_n < -\lambda_{\bar k}(\Omega)$. For any real $t_1,\dots t_{\bar k}$we define\[\phi := \sum_{h=1}^{\bar k} t_h \varphi_h.\]By denoting $J_{\lambda,\mu}(u)={\mathcal{E}}_\mu(u)+\frac{\lambda}{2}\|u\|_{L^2}^2$, so thatMorse index properties can be written in terms of $J''_{\lambda,\mu}$, we have\[\begin{split}J''_{\lambda_n,\mu_n}(u_n)[u_n,\phi] &= -(p-1)\mu_n\int_\Omega |u_n|^{p-1}u_n\phi,\\J''_{\lambda_n,\mu_n}(u_n)[\phi,\phi] &= \sum_{h=1}^{\bar k} t_h^2 \int_\Omega \bigl (|\nabla \varphi_h| + \lambda_n \varphi_h^2\bigr )\,dx - p\mu_n\int_\Omega |u_n|^{p-1}\phi^2\,dx \\ &\leq \sum_{h=1}^{\bar k} t_h^2(\lambda_{h}(\Omega) + \lambda_n) - (p-1)\mu_n\int_\Omega |u_n|^{p-1}\phi^2\,dx \leq - (p-1)\mu_n\int_\Omega |u_n|^{p-1}\phi^2\,dx,\end{split}\]where equality holds if and only if $t_1=\dots=t_{\bar k}=0$. As a consequence\begin{multline*}J''_{\lambda_n,\mu_n}(u_n)[t_0 u_n+ \phi, t_0 u_n + \phi] \leq -t_0^2 (p-1)\mu_n\int_\Omega |u_n|^{p-1}u_n^2 \\- 2t_0(p-1)\mu_n\int_\Omega |u_n|^{p-1}u_n\phi \,dx - (p-1)\mu_n\int_\Omega |u_n|^{p-1}\phi^2\,dx.\end{multline*}We deduce that $J''_{\lambda_n,\mu_n}(u_n)$ is negative definite on $\spann\{u_n, \varphi_1,\dots,\varphi_{\bar k}\}$, in contradiction with the bound on the Morse index (note that$u_n$ cannot be a linear combination of a finite number of eigenfunctions, otherwise using the equations we would obtain that such eigenfunctions are linearly dependent).\end{proof}\begin{lemma}\label{localblow}Let \eqref{eq:mainass_secMorse} hold.Then $\lambda_n\to +\infty$.\end{lemma}\begin{proof}By Lemma \ref{lem:lambda_bdd_below} we have that $\lambda_n$ is bounded below. As a consequence, we can use H\"{o}lder inequality with $\|u_n\|_{L^2}=1$ and \eqref{eq:auxiliary_n} to write\[\mu_n\,\|u_n\|^{p-1}_{L^{\infty}}\ge \mu_n \,\|u_n\|^{p+1}_{L^{p+1}}=\alpha_n+\lambda_n\rightarrow +\infty.\]Let us define\begin{equation}\label{equn}U_n: =\mu_n^{\frac{1}{p-1}}\,u_n,\quad\text{ so that }-\Delta U_n+\lambda_n U_n=|U_n|^{p-1}U_n\quad \text{in}\,\,\Omega,\quadU|_{\partial\Omega}=0.\end{equation}Pick $P_n\in\Omega$ such that $|U_n(P_n)|=\|U_n\|_{L^{\infty}(\Omega)}$ and set\begin{equation}\label{tildepsn}\tilde\varepsilon_n: =|U_n(P_n)|^{-\frac{p-1}{2}}=\frac{1}{\sqrt{\mu_n\,\|u_n\|^{p-1}_{L^{\infty}}}}\longrightarrow 0\end{equation}Hence, $|U_n(P_n)|\to+\infty$; moreover, as $P_n$ is a point of positive maximum or of negative minimum, we have\begin{equation}\nonumber0\le \frac{-\Delta U_n(P_n)}{U_n(P_n)}=|U_n(P_n)|^{p-1}-\lambda_n\,.\end{equation}Thus $\lambda_n|U_n(P_n)|^{1-p}\le 1$, and since $\lambda_n$ is bounded from below, we conclude\begin{equation}\label{limtildelam}\frac{\lambda_n}{|U_n(P_n)|^{p-1}}\longrightarrow \tilde\lambda\in [0,1].\end{equation}Now, we are left to prove that $\tilde\lambda>0$. Let us define\begin{equation}\label{tildeVn}\tilde V_n(y)=\tilde\varepsilon_n^{\frac{2}{p-1}}\, U_n(\tilde\varepsilon_n\,y+P_n),\quad\quad y\in \tilde\Omega_n:=\big (\Omega-P_n\big )/\tilde\varepsilon_n,\end{equation}and let $d_n := d(P_n,\partial\Omega)$; we have, up to subsequences,\[\frac{\tilde\varepsilon_n}{d_n}\longrightarrow L\in [0,+\infty]\qquad\text{and}\qquad\tilde\Omega_n\rightarrow\left\{ \begin{array}{ll} {\mathbb{R}}^n, & \text{if $L=0$;} \\ H, & \text{if $L>0$,} \end{array} \right.\]where $H$ is a half-space such that $0\in \overline H$ and $d(0,\partial H)=1/L$.The function $\tilde V_n$ satisfies\begin{equation}\nonumber\left\{ \begin{array}{ll} -\Delta \tilde V_n+\lambda_n\,\tilde \varepsilon_n^2\,\tilde V_n=|\tilde V_n|^{p-1}\tilde V_n, & \hbox{in}\,\, \tilde\Omega_n;\\ |\tilde V_n|\le |\tilde V_n(0)|=1, & \hbox{in}\,\, \tilde\Omega_n;\\ \tilde V_n=0, & \hbox{on}\,\, \partial\tilde \Omega_n. \end{array}\right.\end{equation}From \eqref{tildepsn} and \eqref{limtildelam} we get $\tilde\varepsilon_n^2\,\lambda_n\rightarrow \tilde\lambda$; hence, by elliptic regularity and up to a further subsequence, $\tilde V_n\rightarrow \tilde V$ in $\mathcal{C}^1_{{\mathrm{loc}}}(\overline H)$ where $\tilde V$ solves\begin{equation}\label{limprob1}\left\{ \begin{array}{ll} -\Delta \tilde V+\tilde\lambda\,\tilde V=|\tilde V|^{p-1}\tilde V, & \hbox{in}\,\, H;\\ |\tilde V|\le |\tilde V(0)|=1, & \hbox{in}\,\, H;\\ \tilde V=0, & \hbox{on}\,\, \partial H. \end{array}\right.\end{equation}Since $\sup_n m(U_n)\leq \bar k$ (as a solution to \eqref{equn}), one can show as in Theorem $3.1$ of \cite{MR2825606} that $m(\tilde V)\leq \bar k$. In particular, $\tilde V$ is stable outside a compact set (see Definition $2.1$ in \cite{MR2825606}) so that, by Theorem $2.3$ and Remark $2.4$ of \cite{MR2825606}, we have$$\tilde V(x)\rightarrow 0 \quad\quad \text{as} \quad\quad |x|\rightarrow +\infty.$$Moreover, since $\tilde V$ is not trivial, we also have that $\tilde\lambda>0$. For, if $\tilde\lambda=0$ the function$\tilde V$ would be a solution of the Lane-Emden equation$-\Delta u=|u|^{p-1}u$ either in ${\mathbb{R}}^n$ or in $H$. In both cases, $\tilde V$ would contradict Theorems $2$ and $9$ of \cite{MR2322150}, being non trivial and stable outside a compact set. Thus, $\tilde\lambda >0$ and by \eqref{limtildelam} we conclude $\lambda_n\rightarrow +\infty$.\end{proof}\begin{remark}\label{rem4}We stress that the scaling argument in Lemma \ref{localblow}, leading to the limit problem \eqref{limprob1} (with $\tilde\lambda>0$), can be repeated also near points of \emph{local} extremum.More precisely, let $Q_n$ be such that $|U_n(Q_n)|\to +\infty$ and$$|U_n(Q_n)|=\max_{\Omega\cap B_{R_n\tilde\varepsilon_n}(Q_n)}U_n,$$for some $R_n\to +\infty$. Then the above procedure can be repeated by replacing $P_n$ with $Q_n$ in definition \eqref{tildepsn}.\end{remark}The local description of the asymptotic behaviour of the solutions $U_n$ to \eqref{equn} with bounded Morse index can be carried out more conveniently by definingthe sequence (see \cite[Theorem $3.1$]{MR2825606})\begin{equation}\label{defVn}V_n(y)=\varepsilon_n^{\frac{2}{p-1}}\, U_n(\varepsilon_n\,y+P_n),\quad y\in \Omega_n:=\frac{\Omega-P_n}{\varepsilon_n},\end{equation}where $P_n$ is defined before \eqref{tildepsn}, and $\varepsilon_n=\frac{1}{\sqrt{\lambda_n}}\to 0$.Then, $V_n$ satisfies\begin{equation}\nonumber\left\{ \begin{array}{ll} -\Delta V_n+ V_n=| V_n|^{p-1} V_n, & \hbox{in}\,\,\Omega_n;\\ |V_n|\le | V_n(0)|=\big ({\varepsilon_n/\tilde\varepsilon_n}\big )^{\frac{2}{p-1}}\rightarrow\tilde\lambda^{-\frac{1}{p-1}}, & \hbox{in}\,\, \Omega_n;\\ V_n=0, & \hbox{on}\,\, \partial\Omega_n. \end{array}\right.\end{equation}As before, we have (up to a subsequence) $V_n\rightarrow V$ in $\mathcal{C}^1_{\mathrm{loc}}(\overline H)$ where $H$ is either ${\mathbb{R}}^N$ or a half space and $V$ solves\begin{equation}\label{limprob2}\left\{ \begin{array}{ll} -\Delta V+ V=| V|^{p-1} V, & \hbox{in}\,\, H;\\ |V|\le | V(0)|=\tilde\lambda^{-\frac{1}{p-1}}, & \hbox{in}\,\, H;\\ V=0, & \hbox{on}\,\, \partial H. \end{array}\right.\end{equation}By recalling the discussion following \eqref{limprob1} we also have $m(V)<+\infty$.We collect some well known property of such a $V$ in the following result.\begin{theorem}[\cite{MR688279,MR2825606,MR2322150,MR2785899}]\label{thm:unif_est_Farina}Let $V$ be a classical solution to \eqref{limprob2} such that $m(V)\leq\bar k$. Then:\begin{enumerate} \item $H={\mathbb{R}}^N$; \item $V(x)\to 0$ as $|x|\rightarrow +\infty$, $V \in H^1({\mathbb{R}}^N)\cap L^{p+1}({\mathbb{R}}^N)$; \item there exist $C$ only depending on $\bar k$ (and not on $V$) such that \[ \|V\|_{L^{\infty}} + \|\nabla V\|_{L^{\infty}}<C. \]\end{enumerate}\end{theorem}\begin{proof}Claim 2 follows from Theorem 2.3 and Remark 2.4 of \cite{MR2825606}, see also \cite[Remark 1.4]{MR688279}. As a consequence,Theorem 1.1 of \cite[Remark 1.4]{MR688279} readily applies, providing claim 1 ($V$ is not trivial as $V(0)>0$). On the other hand, the $L^\infty$ estimates in claim 3.are proved in Theorem 1.9 of \cite{MR2785899}.\end{proof}\begin{corollary}\label{distpnbound}If the sequence $\{U_n\}$ of solutions to \eqref{equn} has uniformly bounded Morse index, and if $P_n\in\Omega$ is such that $|U_n(P_n)|=\|U_n\|_{L^{\infty}(\Omega)}\to+\infty$, then$$\sqrt{\lambda_n}\,d(P_n,\partial\Omega)\rightarrow +\infty,\qquad\text{where }\frac{\lambda_n}{|U_n(P_n)|^{p-1}}\to\tilde\lambda\in(0,1].$$\end{corollary}\begin{remark}Recall that $Z_{N,p}$, the unique positive solution to $-\Delta u+ u=| u|^{p-1} u$ in ${\mathbb{R}}^N$, has Morse index $1$ \cite{MR969899}; then, if $V$ solves \eqref{limprob2} in ${\mathbb{R}}^N$and $1<m(V)<+\infty$, then $V$ is necessarily sign-changing.\end{remark}Following the same pattern as in \cite{MR2825606}, we now analyze the global behaviour of a sequence $\{U_n\}$ of solutions to \eqref{equn} for $\lambda_n\to +\infty$, assuming that\(\lim_{n\to +\infty} m(U_n)\leq\bar k<\infty.\)By the previous discussion, if $P^1_n$ is a sequence of points such that $|U_n(P^1_n)|=\|U_n\|_{L^{\infty}(\Omega)}$, we have$|U_n(P^1_n)|\rightarrow +\infty$ and ${\lambda_n}\,d(P^1_n,\partial\Omega)^2\rightarrow +\infty$. We now look for other possible sequences of (local) extremum points $P^i_n$, $i=2,3,..$, along which $|U_n|$ goes to infinity. For any $R>0$, consider the quantity\begin{equation}\nonumberh_1(R)=\limsup_{n\to +\infty} \Bigl (\lambda_n^{-\frac{1}{p-1}}\max_{|x-P^1_n|\ge R\,\lambda_n^{-1/2}}|U_n(x)| \Bigr ).\end{equation}We will prove that if $h_1(R)$ is \emph{not vanishing} for large $R$, then there exists a 'blow-up' sequence $P^2_n$ for $u_n$, 'disjoint' from $P^1_n$. Indeed, let us suppose that\begin{equation}\nonumber\limsup_{R\to +\infty} h_1(R)=4\delta>0.\end{equation}Hence, up to a subsequence and for arbitrarily large $R$, we have\begin{equation}\label{ass1}\lambda_n^{-\frac{1}{p-1}}\max_{|x-P^1_n|\ge R\,\lambda_n^{-1/2}}|U_n(x)| \ge 2\delta.\end{equation}Since $U_n$ vanishes on $\partial\Omega$, there exists$P^2_n\in\Omega\backslash B_{R\,\lambda_n^{-1/2}}(P_n^1)$such that\begin{equation}\label{pn2}|U_n(P_n^2)|=\max_{|x-P^1_n|\ge R\,\lambda_n^{-1/2}}|U_n(x)|.\end{equation}Clearly, assumption \eqref{ass1} implies that $|U_n(P_n^2)|\rightarrow +\infty$. We first prove that the sequences $P_n^1$ and $P_n^2$ are far away each other.\begin{lemma}\label{disj}Take $R$ such that \eqref{ass1} holds, and let $P_n^2$ be defined as in \eqref{pn2}; then\begin{equation}\label{limp1p2}\lambda_n^{1/2}|P_n^2-P^1_n|\rightarrow +\infty\end{equation}as $n\to \infty$.\end{lemma}\begin{proof}Assuming the contrary one would get, up to a subsequence\[\lambda_n^{1/2}|P_n^2-P^1_n|\rightarrow R'\ge R.\]Let us now recallthat by \eqref{defVn} and the subsequent discussion, we have:\begin{equation}\label{limblowseq}\lambda_n^{-\frac{1}{p-1}}\, U_n(\lambda_n^{-1/2}\,y+P^1_n) =: V^1_n(y)\rightarrow V(y)\quad \textrm{in}\,\, \mathcal{C}^1_{{\mathrm{loc}}}({\mathbb{R}}^N)\end{equation}as $n\to +\infty$. Then, up to subsequences,\begin{equation}\nonumber\lambda_n^{-\frac{1}{p-1}}\,|U_n(P_n^2)|=\big |V^1_n\bigr(\lambda_n^{1/2}(P_n^2-P^1_n)\bigl)\big|\rightarrow \big |V(y')\big |,\quad |y'|=R'\ge R.\end{equation}Since $V$ is vanishing for $|y|\to +\infty$, one can choose $R$ such that$|V(y)|\le\delta$ for every $ |y|\ge R$.But this contradicts \eqref{ass1}.\end{proof}Furthermore, we also have that the blow-up points stay far away from the boundary.\begin{lemma}\label{distbd}Assume \eqref{ass1} and let $P_n^2$ be defined as in \eqref{pn2}; then\begin{equation}\label{distp2nbound}\sqrt{\lambda_n}\,d(P^2_n,\partial\Omega)\rightarrow +\infty\end{equation}as $n\to \infty$. Moreover,\begin{equation}\label{maxp2nball}|U_n(P_n^2)|=\max_{\Omega\cap B_{R_n\lambda^{-1/2}_n}(P^2_n)}|U_n|\end{equation}for some $R_n\to +\infty$.\end{lemma}\begin{proof}Let us set\begin{equation}\nonumber\tilde\varepsilon^2_n: =|U_n(P^2_n)|^{-\frac{p-1}{2}}\quad \mathrm{and}\quad R_n^{(2)}:=\frac{1}{2}\,\frac{|P_n^2-P_n^1|}{\tilde\varepsilon^2_n}.\end{equation}Clearly, $\tilde\varepsilon^2_n\rightarrow 0$; moreover, by \eqref{ass1} and \eqref{pn2},$\tilde\varepsilon^2_n\le (2\delta)^{-\frac{p-1}{2}}\lambda_n^{-1/2}$, so that$$R^{(2)}_n\ge \frac{(2\delta)^{\frac{p-1}{2}}}{2}\,\lambda_n^{1/2}\,{|P_n^2-P_n^1|}\rightarrow +\infty, $$as $n\to +\infty$ by Lemma \ref{disj}. We claim that this implies\begin{equation}\label{maxp2n}|U_n(P_n^2)|=\max_{\Omega\cap B_{R^{(2)}_n\tilde\varepsilon^2_n}(P^2_n)}| U_n|.\end{equation}For, if $x\in B_{R^{(2)}_n\tilde\varepsilon^2_n}(P^2_n)$, by \eqref{limp1p2} we would have$$|x-P^1_n|\ge |P^2_n-P^1_n|-|x-P^2_n|\ge \frac{1}{2}\,|P^2_n-P^1_n|\ge R\,\lambda_n^{-1/2},$$for arbitrarily large $R$. This means that$$\Omega\cap B_{R^{(2)}_n\tilde\varepsilon^2_n}(P^2_n)\subset \Omega\backslash B_{R\,\lambda_n^{-1/2}}(P^1_n).$$Then, the claim follows. Now, by recalling Remark \ref{rem4}, we can apply to $U_n$ satisfying \eqref{maxp2n} the same scaling arguments as in the proof of Lemma \ref{localblow}, so that we conclude$$0< \lim_{n\to +\infty}\tilde\varepsilon_n^2\,\sqrt{\lambda_n}.$$Hence, \eqref{maxp2nball} holds by defining $R_n=R^{(2)}_n\tilde\varepsilon_n^2\,\sqrt{\lambda_n}$,and \eqref{distp2nbound} follows by Corollary \ref{distpnbound}.\end{proof}We can now iterate the previous arguments: let us define, for $k\ge 1$,\begin{equation}\label{defhn}h_k(R)=\limsup_{n\to +\infty} \Bigl (\lambda_n^{-\frac{1}{p-1}}\max_{d_{n,k}(x)\ge R\,\lambda_n^{-1/2}}|U_n(x)| \Bigr ),\end{equation}where$$d_{n,k}(x): =\min\{|x-P^i_n|\,:\, i=1,...,k\}$$and the sequences $P^i_n$ are such that\begin{equation}\nonumber\sqrt{\lambda_n}\,d(P^i_n,\partial\Omega)\rightarrow +\infty;\quad \lambda_n^{1/2}|P_n^i-P^j_n|\rightarrow +\infty,\quad\quad i,j=1,...,k,\quad i\neq j\end{equation}as $n\to +\infty$.Assume that$$\limsup_{n\to +\infty} h_k(R)=4\delta>0.$$As before, up to a subsequence and for arbitrarily large $R$, we have\begin{equation}\label{assk}\lambda_n^{-\frac{1}{p-1}}\max_{d_{n,k}(x)\ge R\,\lambda_n^{-1/2}}|U_n(x)| \ge 2\delta\end{equation}and there exist $P^{k+1}_n$ so that\begin{equation}\nonumber|U_n(P_n^{k+1})|=\max_{d_{n,k}(x)\ge R\,\lambda_n^{-1/2}}|U_n(x)|\end{equation}with $\lim_{n\to +\infty}|U_n(P_n^{k+1})|=+\infty$.Moreover, as in Lemma \ref{disj} we deduce that, for every $i=1,...,k$\begin{equation}\label{limblowseqi}\lambda_n^{-\frac{1}{p-1}}\,U_n(\lambda_n^{-1/2}\,y+P^i_n): = V^i_n(y)\rightarrow V^i(y)\quad \textrm{in}\,\, \mathcal{C}^1_{{\mathrm{loc}}}({\mathbb{R}}^N)\end{equation}as $n\to +\infty$; hence, by \eqref{assk} and again from the vanishing of $V$ at infinity, we conclude that\begin{equation}\lambda_n^{1/2}|P_n^{k+1}-P^i_n|\rightarrow +\infty\end{equation}as $n\to \infty$, for every $i=1,...,k$. Setting now\begin{equation}\nonumber\tilde\varepsilon^{k+1}_n: =|U_n(P^{k+1}_n)|^{-\frac{p-1}{2}}\quad \mathrm{and}\quad R_n^{(k+1)}:=\frac{1}{2}\,\frac{d_{n,k}(P^{k+1}_n)}{\tilde\varepsilon^{k+1}_n}\end{equation}we still have $\tilde\varepsilon^{k+1}_n\to 0$ and, by \eqref{assk}, $R_n^{(k+1)} \to +\infty$ as $n\to \infty$ (see Lemma \ref{distbd}). Then, by the same arguments as in Lemma \ref{distbd}, we get\begin{equation}\label{maxpkn}|U_n(P_n^{k+1})|=\max_{\Omega\cap B_{R^{(k+1)}_n\tilde\varepsilon^{k+1}_n}(P^{k+1}_n)} |u_n|\,,\end{equation}and furthermore$$ \lim_{n\to +\infty}\tilde\varepsilon_n^{k+1}\,\sqrt{\lambda_n}>0\,,$$so that by defining $R_n=:R^{(k+1)}_n\tilde\varepsilon_n^{k+1}\,\sqrt{\lambda_n}\rightarrow +\infty$we have\begin{equation}\label{maxpknball}|U_n(P_n^{k+1})|=\max_{\Omega\cap B_{R_n\lambda^{-1/2}_n}(P^{k+1}_n)}| U_n|.\end{equation}Now, by the same arguments as in \cite{MR2825606}, it turns out that the iterative procedure must stop after \emph{at most} $\bar k-1$ steps, where$\bar k =\lim_{n\to +\infty} m(u_n)$. Thus, we have proved:\begin{proposition}\label{glob1}Let $\{U_n\}_n$ be a solution sequence to \eqref{equn} such that $\lambda_n\to+\infty$and $m(U_n)\leq\bar k$.Then, up to a subsequence, there exist $P_n^1,...,P_n^k$, with $k\le \bar k$ such that\begin{equation}\label{limpin}\sqrt{\lambda_n}\,d(P^i_n,\partial\Omega)\rightarrow +\infty;\quad \lambda_n^{1/2}|P_n^i-P^j_n|\rightarrow +\infty,\quad\quad i,j=1,...,k,\quad i\neq j\end{equation}as $n\to +\infty$ and\begin{equation}\nonumber|U_n(P_n^{i})|=\max_{\Omega\cap B_{R_n\lambda^{-1/2}_n}(P^{i}_n)}|U_n|,\quad i=1,...,k,\end{equation}for some $R_n\to +\infty$ as $n\to +\infty$. Finally,\begin{equation}\label{limhr0}\lim_{R\to +\infty} h_k(R)=0\end{equation}where $h_k(R)$ is given by \eqref{defhn}.\end{proposition}We now show that the sequence $U_n$ decays exponentially away from the blow-up points.\begin{proposition}\label{glob2}Let $\{U_n\}_n$ satisfy the assumptions of Proposition \ref{glob1}. Then, there exist$P_n^1,...,P_n^k$ andpositive constants $C$, $\gamma$, such that\begin{equation}\label{stimglob}|U_n(x)|\le C\lambda^{\frac{1}{p-1}}_n \sum_{i=1}^ke^{-\gamma\sqrt{\lambda_n}|x-P_n^i|}\,,\quad\quad \forall \,x\in\Omega,\quad n\in{\mathbb{N}}\,.\end{equation}\end{proposition}\begin{proof}By \eqref{limhr0}, for large $R>0$ and $n>n_0(R)$ it holds\begin{equation}\nonumber\lambda_n^{-\frac{1}{p-1}}\max_{d_{n,k}(x)\ge R\,\lambda_n^{-1/2}}|U_n(x)| \le \Bigr (\frac{1}{2p} \Bigl )^{\frac{1}{p-1} }\end{equation}Then, for $n>n_0(R)$ and for $x\in \{d_{n,k}(x)\ge R\,\lambda_n^{-1/2}\}$, we have\begin{equation}\nonumbera_n(x): = \lambda_n-p |U_n(x)|^{p-1}\ge \lambda_n-\frac{\lambda_n}{2}=\frac{\lambda_n}{2}\end{equation}We stress that the linear operator\begin{equation}\nonumberL_n: =-\Delta + a_n(x)\end{equation}comes from the linearization of equation \eqref{equn} at $U_n$; let us compute this operator on the functions$$\phi^i_n(x)=e^{-\gamma\sqrt{\lambda_n}\,|x-P^i_n|}\,,\quad\quad \gamma>0,\quad\quad i=1,...,k$$in $\{d_{n,k}(x)\ge R\,\lambda_n^{-1/2}\}$. We obtain:$$L_n \phi^i_n(x)=\lambda_n\phi^i_n(x)\Bigr [-\gamma^2+(N-1)\frac{\gamma}{\sqrt{\lambda_n}\,|x-P^i_n|}+\frac{a_n(x)}{\lambda_n}\Bigl ]\ge \lambda_n\phi^i_n(x)\bigr [-\gamma^2+1/2\bigl ]\ge 0$$for $n$ large, provided $0<\gamma\le 1/\sqrt 2$. Moreover, for $|x-P^i_n|=R\lambda_n^{-1/2}$, $ i=1,...,k,$ and $R$ large we have$$e^{\gamma R}\phi^i_n(x)-\lambda_n^{-\frac{1}{p-1}}|U_n(x)|=1-\lambda_n^{-\frac{1}{p-1}}|U_n(x)|>0$$as $n\to +\infty$, by \eqref{limblowseq}. Note further that$$\{x:d_{n,k}(x)= R\,\lambda_n^{-1/2}\} = \bigcup_{i=1}^k \partial B_{R\,\lambda_n^{-1/2}}(P_n^i)\subset \Omega$$for large enough $n$. Then, by defining\begin{equation}\nonumber\phi_n: = e^{\gamma R}\lambda_n^{\frac{1}{p-1}}\sum_{i=1}^k \,\phi^i_n\end{equation}we have$$\phi_n(x)-|U_n(x)|\ge 0\quad\quad \mathrm{on}\quad\quad\{d_{n,k}(x)= R\,\lambda_n^{-1/2}\}\cup\partial\Omega$$and\begin{equation}\nonumberL_n(\phi_n-|U_n|)\ge -L_n\,|U_n|=\Delta \,|U_n|-\lambda_n\,|U_n|+p|U_n|^p\ge (p-1)\,|U_n|^p\ge 0\end{equation}in $\Omega\backslash \{d_{n,k}(x)\le R\,\lambda_n^{-1/2}\}$. Then (for $R$ large and $n\ge n_0(R)$)we obtain $|U_n|\le \phi_n$ in the same set, by the minimum principle. Moreover, since by \eqref{limtildelam}$$|U_n(x)|\le\|U_n\|_{L^{\infty}(\Omega)}=|U_n(P^1_n)|\le C \lambda_n^{\frac{1}{p-1}}$$for some $C>0$, we also have, in $\{d_{n,k}(x)\le R\,\lambda_n^{-1/2}\}$,$$|U_n(x)|\le\|U_n(x)\|_{L^{\infty}(\Omega)}=|U_n(P^1_n)|\le C e^{\gamma R}\lambda_n^{\frac{1}{p-1}}\sum_{i=1}^k e^{-\gamma\sqrt{\lambda_n}|x-P_n^i|}.$$Then, possibly by choosing a larger $C$, estimate \eqref{stimglob} follows for every $n$.\end{proof}We now exploit the previous results to show that suitable rescalings of the solutions to \eqref{eq:auxiliary_n} converge (locally) to some bounded solution $V$ of\begin{equation}\label{eqV}-\Delta V+ V=| V|^{p-1} V\end{equation}in ${\mathbb{R}}^N$.\begin{lemma}\label{lemlim1}Let \eqref{eq:mainass_secMorse} hold.Then $|u_n|$ admits $k\le \bar k$ local maxima $P_n^1,...,P_n^k$ in $\Omega$ such that, defining\begin{equation}u_{i,n}(x)= \Bigl ( \frac{\mu_n}{\lambda_n}\Bigr )^{\frac{1}{p-1}}u_n \bigr (\frac{x}{\sqrt {\lambda_n}}+P_n^i\bigl ),\quad\quad x\in \Omega_{n,i}:=\sqrt{\lambda_n}\bigr (\Omega-P_n^i \bigl ),\end{equation}it results, up to a subsequence,\begin{equation}u_{i,n}(x)\rightarrow V_i\quad\quad \mathrm{in}\,\,\mathcal{C}^1_{{\mathrm{loc}}}({\mathbb{R}}^n)\quad \mathrm{as}\,\,n\to +\infty,\quad \forall\,\,i=1,2,...,k,\end{equation}where $V_i$ is a bounded solution of \eqref{eqV} with $m(V_i)\le \bar{k}$.\noindent As a consequence, for every $q\ge 1$,\begin{equation}\label{convlq}\Bigl ( \frac{\mu_n}{\lambda_n}\Bigr )^{\frac{q}{p-1}}\lambda_n^{N/2}\int_{\Omega} |u_n|^q \,dx\rightarrow\sum_{i=1}^{k}\int_{{\mathbb{R}}^n}|V_i|^q\,dx\quad\quad \mathrm{as}\,\, n\to +\infty.\end{equation}\end{lemma}\begin{proof}By Lemma \ref{localblow} we have $\lambda_n\to +\infty$; then, the first part of the lemma follows by definition \eqref{equn}, by \eqref{limblowseqi} and by Proposition \ref{glob1}; by the same propositionand by Proposition \ref{glob2} we also have that the local maxima $P^i_n$ satisfies \eqref{limpin} and that the pointwise estimate\begin{equation}\label{stimglobvn}|u_n(x)|\le C\Bigl (\frac{\lambda_n}{\mu_n}\Bigr )^{\frac{1}{p-1}} \sum_{i=1}^ke^{-\gamma\sqrt{\lambda_n}|x-P_n^i|}\,,\quad\quad \forall \,x\in\Omega,\quad n\in{\mathbb{N}}\,.\end{equation}holds. Let us fix $R>0$ and set $r_n=R/\sqrt{\lambda_n}$; for large enough $n$, \eqref{limpin} implies$$B_{r_n}(P^i_n)\subset \Omega,\quad\quad B_{r_n}(P^i_n)\cap B_{r_n}(P^j_n)=\emptyset, \quad i\neq j.$$Then we obtain\[\begin{array}{cl}&\left |\left ( \frac{\mu_n}{\lambda_n}\right)^{\frac{q}{p-1}}\lambda_n^{N/2}\int_{\Omega} |u_n|^q \,dx-\sum_{j=1}^{k}\int_{B_R(0)}|u_{j,n}|^q\,dx\,\,\right |\smallskip\\&=\left ( \frac{\mu_n}{\lambda_n}\right )^{\frac{q}{p-1}}\lambda_n^{N/2}\left |\int_{\Omega} |u_n|^q \,dx-\sum_{j=1}^{k}\int_{B_{r_n}(P^j_n)}|u_{n}|^q\,dx\,\,\right |\smallskip\\&=\left ( \frac{\mu_n}{\lambda_n}\right )^{\frac{q}{p-1}}\lambda_n^{N/2}\int_{\Omega\backslash \bigcup_{j=1}^k\,B_{r_n}(P^j_n)} |u_n|^q \,dx\le C^q\lambda_n^{N/2}\int_{\Omega\backslash \bigcup_{j=1}^k\,B_{r_n}(P^j_n)} \left |\sum_{i=1}^ke^{-\gamma\sqrt{\lambda_n}|x-P_n^i|}\right |^q \,dx\smallskip\\&\le C^qk^{q-1}\lambda_n^{N/2} \sum_{i=1}^k\int_{\Omega\backslash \bigcup_{j=1}^k\,B_{r_n}(P^j_n)} e^{-q\gamma\sqrt{\lambda_n}|x-P_n^i|} \,dx\smallskip\\&\le C^qk^{q-1}\lambda_n^{N/2} \sum_{i=1}^k\int_{{\mathbb{R}}^N\backslash \,B_{r_n}(P^i_n)} e^{-q\gamma\sqrt{\lambda_n}|x-P_n^i|} \,dx\smallskip\\&\le (Ck)^{q}\sum_{i=1}^k\int_{{\mathbb{R}}^N\backslash \,B_{R}(0)} e^{-q\gamma\,|y|} \,dy\le C_1 \,e^{-C_2 R},\end{array}\]for some positive $C_1$, $C_2$. Letting $n\to +\infty$ we have, up to subsequences,\begin{multline*}\Bigg |\lim_{n\to +\infty}\Bigl ( \frac{\mu_n}{\lambda_n}\Bigr )^{\frac{q}{p-1}}\lambda_n^{N/2}\int_{\Omega} |u_n|^q \,dx-\sum_{i=1}^{k}\int_{B_R(0)}|V_i|^q\,dx\,\,\Bigg |\\=\lim_{n\to +\infty}\Bigg |\Bigl ( \frac{\mu_n}{\lambda_n}\Bigr )^{\frac{q}{p-1}}\lambda_n^{N/2}\int_{\Omega} |u_n|^q \,dx-\sum_{i=1}^{k}\int_{B_R(0)}|u_{i,n}|^q\,dx\,\,\Bigg |\le C_1 \,e^{-C_2 R}.\end{multline*}Then, \eqref{convlq} follows by taking $R\to +\infty$.\end{proof}The previous lemma allows us to gain some information on the asymptotic behavior of the sequences $\lambda_n$, $\mu_n$ and $\|u_n\|_{L^{p+1}(\Omega)}$. We first provide some bounds for the solutions of the limit problem \eqref{eqV} which will be useful in the sequel.\begin{lemma}\label{boundbelow}Let $V_i$, $i=1,\dots,k$ be as in Lemma \ref{lemlim1} (so that $m(V_i)\leq\bar k$). There exists a constant $C$, only depending on the full sequence $\{u_n\}_n$ and not on $V_i$(and on the particular associated subsequence), such that\[\|V_i\|_{H^1}^2 = \|V_i\|_{L^{p+1}}^{p+1} \leq C.\]Furthermore, if also $m(V_i)\geq2$ (or, equivalently, if $V_i$ changes sign)the following estimates hold:\begin{equation}\label{uppstiml2}\|V_i\|^{p+1}_{L^{p+1}}> 2\,\|Z\|^{{p+1}}_{L^{{p+1}}},\qquad\|V_i\|^2_{L^2}> 2\,\|Z\|^{2}_{L^{2}},\end{equation}where $Z\equiv Z_{N,p}$ is the unique positive solution to \eqref{eqV}.\end{lemma}\begin{proof}To prove the bounds from above we claim that there exists $\bar R>0$, not depending on $i$, such that $V_i$ is stable outside $\overline{B_{\bar R}}$. Thenthe desired estimate will follow, since\[\|V_i\|^{p+1}_{L^{p+1}} = \int_{B_{\bar R}} |V_i|^{p+1} + \int_{{\mathbb{R}}^N\setminus B_{\bar R}} |V_i|^{p+1},\]where the first term is uniformly bounded by Theorem \ref{thm:unif_est_Farina}, while the second one can be estimated in an uniform wayby reasoning as in the proof of \cite[Theorem 2.3]{MR2825606}. To prove the claim, recalling \eqref{defhn} and \eqref{limhr0}, let$\bar R$ be such that\[h_k(\bar R) \leq \left(\frac{1}{p}\right)^{1/(p-1)}.\]Then $|V_i(x)|^{p-1}\leq 1/p $ on ${\mathbb{R}}^N\setminus B_{\bar R}$ and thus, for any $\psi\in C^\infty_0({\mathbb{R}}^N)$, $\psi\equiv0$ in $B_{\bar R}$, it holds\[\int_{{\mathbb{R}}^N} |\nabla \psi|^2 + \psi^2 - p|V_i|^{p-1}\psi^2\,dx \geq \left( 1 - p \|V_i\|^{p-1}_{L^{\infty}({\mathbb{R}}^N\setminus B_{\bar R})}\right)\int_{{\mathbb{R}}^N} \psi^2 \geq 0.\]Hence $V_i$ is stable outside $B_{\bar R}$, and the first part of the lemma follows.On the other hand, if $V_i$ is a sign-changing solution to \eqref{eqV}, the associated energy functional\begin{equation}\nonumberE(V_i)= \frac{1}{2}\|\nabla V_i\|^2_{L^2}+\frac{1}{2}\|V_i\|^2_{L^2}-\frac{1}{p+1}\|V_i\|^{p+1}_{L^{p+1}}\end{equation} satisfies the following \emph{energy doubling property} (see \cite{MR2263672}): $$E(V_i)>2\,E(Z)$$ On the other hand, by using the equation $E'(V_i)V_i=0$ and the Pohozaev identity one gets \begin{equation}\label{eulp}\|V_i\|^{p+1}_{L^{p+1}}= 2\,\frac{p+1}{p-1}\,E(V_i),\qquad\|V_i\|^2_{L^2}= \frac{N+2-p\,(N-2)}{p-1}\,E(V_i)\end{equation}Since the ground state solution $Z$ satisfies the same identities, the bounds \eqref{uppstiml2} are readily verified.\end{proof}\begin{proposition}Let \eqref{eq:mainass_secMorse} hold and the functions $V_i$ be defined as in Lemma\ref{lemlim1}. We have, as $n\to +\infty$,\begin{eqnarray}\label{convl2}{\mu_n}^{\frac{2}{p-1}}\,\lambda_n^{N/2-2/(p-1)}&\longrightarrow\sum_{i=1}^{k}\int_{{\mathbb{R}}^n}|V_i|^2\,dx\\\label{convlp}{\mu_n}^{\frac{p+1}{p-1}}\,\lambda_n^{N/2-(p+1)/(p-1)}\int_{\Omega} |u_n|^{p+1} \,dx&\longrightarrow\sum_{i=1}^{k}\int_{{\mathbb{R}}^n}|V_i|^{p+1}\,dx\\\label{convl2grad}\alpha_n\,{\mu_n}^{\frac{2}{p-1}}\,\lambda_n^{N/2-(p+1)/(p-1)}&\longrightarrow\sum_{i=1}^{k}\int_{{\mathbb{R}}^n}|\nabla V_i|^2\,dx.\end{eqnarray}\end{proposition}\begin{proof}The limits \eqref{convl2} and \eqref{convlp} follow respectively by choosing $q=2$ and $q=p+1$ in \eqref{convlq} (recall that $\|u_n\|_{L^{2}}=1$). Furthermore, from the equationsfor $u_n$ and $V_k$, we have\[\alpha_n+\lambda_n=\mu_n\|u_n\|_{L^{p+1}}^{p+1},\qquad\int_{{\mathbb{R}}^n}|\nabla V_i|^2\,dx + \int_{{\mathbb{R}}^n}|V_i|^2\,dx = \int_{{\mathbb{R}}^n}|V_i|^{p+1}\,dx,\]and also \eqref{convl2grad} follows.\end{proof}\begin{corollary}\label{limmass}With the same assumptions as above, we have that\begin{enumerate} \item if $1<p<1+\frac{4}{N}$, then $\mu_n\to +\infty$ \item if $p=1+\frac{4}{N}$, then $\mu_n\to \big (\sum_{i=1}^{k}\|V_i\|_{L^2}^2\big )^{2/N}\ge k^{2/N} \|Z\|_{L^2}^{4/N}$ \item if $1+\frac{4}{N}<p<2^*-1$, then $\mu_n\to 0$.\end{enumerate}Furthermore\begin{equation}\label{limalphalam}\frac{\alpha_n}{\lambda_n}\longrightarrow \frac{N(p-1)}{N+2-p(N-2)}.\end{equation}\end{corollary}\begin{proof}The limits of $\mu_n$ follow by the previous proposition. To prove the lower bound in $2$, recall that either $V_i=Z$ or $V_i$ satisfies \eqref{uppstiml2}. Finally, taking the quotient between \eqref{convl2grad} and \eqref{convl2}, we have$$\frac{\alpha_n}{\lambda_n}\longrightarrow\frac{\sum_{i=1}^{k}\int_{{\mathbb{R}}^n}|\nabla V_i|^2\,dx}{\sum_{i=1}^{k}\int_{{\mathbb{R}}^n}| V_i|^2\,dx}$$On the other hand, for every $i=1,2,...,k$ it holds$$\|\nabla V_i\|_{L^2}^2=\Bigg (\frac{\| V_i\|_{L^{p+1}}^{p+1}}{\|V_i\|_{L^2}^2} -1 \Bigg )\|V_i\|_{L^2}^2=\frac{N(p-1)}{N+2-p(N-2)}\, \|V_i\|_{L^2}^2$$where the last equality follows by \eqref{eulp}. By inserting this into the above limit, we get \eqref{limalphalam}.\end{proof}\begin{proof}[Proof of Theorem \ref{thm:bbd_index}]Let $(U_n,\lambda_n)$ solve \eqref{eq:main_prob_U}, with $\rho=\rho_n\to +\infty$and $m(U_n)\leq k$. Changing variables as in \eqref{eq:main_prob_u}, we have that$u_n=\rho_n^{-1/2}U_n$ satisfies \eqref{eq:auxiliary_n} with $\mu_n = \rho_n^{(p-1)/2} \to+\infty$. As a consequence, Lemma \ref{lemma:case_alpha_n_bounded} guarantees that$\alpha_n\to+\infty$, and Corollary \ref{limmass} yields $p<1+4/N$.On the other hand, by direct minimization of the energy one can show that, if $p<1+4/N$, forevery $\rho>0$ there exists a solution of \eqref{eq:main_prob_U} having Morse index one (seealso Section \ref{sec:1const}).\end{proof}\begin{remark}\label{limGN}Reasoning as above we can also show that\begin{equation}\label{newcnp1}\frac{\int_{\Omega} |u_n|^{p+1} \,dx}{\alpha_n^{N(p-1)/4}}\longrightarrowC_{N,p}\,\frac{\|Z\|_{L^2}^{p-1}}{\big (\sum_{i=1}^{k}\| V_i\|_{L^2}^2\big )^{(p-1)/2}}.\end{equation}\end{remark}\section{Max-min principles with two constraints}\label{sec:2const}In this section we deal with the maximization problem with two constraints introduced in \cite{MR3318740}, aiming at considering more general max-min classes of critical points.Let ${\mathcal{M}}$ be defined in \eqref{Emu} and, for any fixed $\alpha>\lambda_1(\Omega)$, let $\mathcal{B}_\alpha$, $\mathcal{U}_\alpha$ be defined as in \eqref{eq:defBU}. We will look for critical points of the $\mathcal{C}^2$ functional\[f(u)=\int_{\Omega}|u|^{p+1},\quad\quad\quad u\in {\mathcal{M}},\]constrained to $\mathcal{U}_\alpha$. To start with, we notice that the topologicalproperties of such set depend on $\alpha$.\begin{lemma}\label{lemma:tilde_U_manifold}Let $\alpha>\lambda_1(\Omega)$. Then the set\[{\mathcal{U}}_{\alpha}\setminus\left\{ \varphi\in {\mathcal{U}}_\alpha :-\Delta\varphi = \alpha\varphi\right\}\]is a smooth submanifold of $H^1_0(\Omega)$ of codimension 2. In particular, this propertyholds true for ${\mathcal{U}}_\alpha$ itself, provided $\alpha\neq\lambda_k(\Omega)$, for every $k$.\end{lemma}\begin{proof}Let us set $F(u)=(\int_\Omega u^2\,dx-1, \ \int_\Omega|\nabla u|^2\,dx)$. For every$u\in{\mathcal{U}}_\alpha$, if the range of $F'(u)$ is ${\mathbb{R}}^2$ then ${\mathcal{U}}_\alpha$ is a smooth manifoldat $u$. Since\[F'(u)[v]=2\left(\int_\Omega uv\,dx, \ \int_\Omega\nabla u\cdot\nabla v\,dx\right),\qquad\text{for every }v\in H^1_0(\Omega),\]and $F'(u)[u]=2(1,\alpha)$, we have that $F'(u)$ is not surjective if and only if\[\int_\Omega\nabla u\cdot\nabla v\,dx = \alpha \int_\Omega uv\,dx\qquad\text{for every }v\in H^1_0(\Omega). \qedhere\]\end{proof}\begin{remark}If $\varphi$ belongs to the eigenspace corresponding to $\lambda_k(\Omega)$, then$\varphi \in {\mathcal{U}}_{\lambda_k(\Omega)}$. As a consequence ${\mathcal{U}}_{\lambda_k(\Omega)}$ may not be smoothnear $\varphi$. For instance, ${\mathcal{U}}_{\lambda_1(\Omega)}$ consists of two isolated points, $\pm\varphi_1$.\end{remark}Of course ${\mathcal{U}}_\alpha$ is closed and odd, for any $\alpha$. Recalling Definition\ref{def:genus} we deduce that its genus$\gamma({\mathcal{U}}_\alpha)$ is well defined.\begin{lemma}If $\alpha<\lambda_{k+1}(\Omega)$, for some $k$, then $\gamma({\mathcal{U}}_\alpha)\leq k$.\end{lemma}\begin{proof}Let $V_k:=\spann\{\varphi_1,\dots,\varphi_k\}$.Since\[\min\left\{\int_\Omega |\nabla u|^2\,dx : u\in V_k^\perp,\,\int_\Omega u^2\,dx=1\right\}=\lambda_{k+1}(\Omega),\]we have that ${\mathcal{U}} \cap V_k^\perp = \emptyset$, thus the projection\[g := \proj_{V_k} \colon {\mathcal{U}}_\alpha \to V_k\setminus\{0\}\]is a continuous odd map of ${\mathcal{U}}_\alpha$ into $V_k\setminus\{0\}$. Now, let $h\colon{\mathbb{S}}^{m}\to {\mathcal{U}}$ be continuous and odd.Then $g\circ h$ is continuous and odd from ${\mathbb{S}}^{m}$ to $V_k\setminus\{0\}$, and Borsuk-Ulam's Theorem forces $m\leq k-1$.\end{proof}\begin{lemma}\label{lemma:genusbigger}If $\alpha>\lambda_{k}(\Omega)$, for some $k$, then $\gamma({\mathcal{U}}_\alpha)\geq k$.\end{lemma}\begin{proof}To prove the lemma we will construct a continuous map $h\colon {\mathbb{S}}^{k-1} \to {\mathcal{U}}$. Let$\ell\in{\mathbb{N}}$ be such that $\lambda_{\ell+1}(\Omega)>\alpha$. For every $i=1,\dots,k$ we define thefunctions\[u_i:=\left(\frac{\lambda_{\ell+i}(\Omega)-\alpha}{\lambda_{\ell+i}(\Omega)-\lambda_i(\Omega)}\right)^{1/2}\varphi_i+\left(\frac{\alpha-\lambda_{i}(\Omega)}{\lambda_{\ell+i}(\Omega)-\lambda_i(\Omega)}\right)^{1/2}\varphi_{\ell+i}.\]We obtain the following straightforward consequences:\begin{enumerate} \item as $\lambda_i(\Omega)<\alpha<\lambda_{\ell+i}(\Omega)$, for every $i$, $u_i$ is well defined; \item $\int_\Omega u_i^2\,dx=1$, $\int_\Omega |\nabla u_i|^2\,dx=\alpha$; \item for every $j\neq i$ it holds $\int_\Omega u_iu_j\,dx=\int_\Omega \nabla u_i\cdot\nabla u_j \,dx=0$.\end{enumerate}Therefore the map $h\colon {\mathbb{S}}^{k-1} \to {\mathcal{U}}$ defined as\[h\colon x=(x_1,\dots,x_k) \mapsto \sum_{i=1}^k x_iu_i\]has the required properties.\end{proof}Now we turn to the properties of the functional $f$. To start with, it satisfiesthe Palais-Smale (P.S. for short) condition on $\overline{\mathcal{B}}_{\alpha}$; more precisely, the followingholds.\begin{lemma}\label{psball}Every P.S. sequence $u_n$ for $f\big |_{\overline{\mathcal{B}}_{\alpha}}$ is a P.S.sequence for $f\big |_{\mathcal{U}_{\alpha}}$ and has a strongly convergentsubsequence in $\mathcal{U}_{\alpha}$.\end{lemma}\begin{proof}We first show that there are no P.S. sequences in ${\mathcal{B}_{\alpha}}$. In fact, if $u_n$ is such a sequence, there is a sequence of real numbers $k_n$ such that\begin{equation}\label{ps}\int_{\Omega}|u_n|^{p-1}u_n\,v-k_n\int_{\Omega}u_n\,v=o(1)\,\|v\|_{H^1_0}\end{equation}for every $v\in H^1_0(\Omega)$. Since $u_n$ is bounded in $H^1_0(\Omega)$, there is a subsequence (still denoted by $u_n$)weakly convergent to $u\in H^1_0(\Omega)$; moreover, $u_n$ converges strongly in $L^{p+1}(\Omega)$ and in $L^2(\Omega)$ to the same limit. By choosing $v=u_n$, we see that $k_n$ is bounded, so that we can also assume that $k_n\rightarrow k$. By taking the limit of \eqref{ps} for $n\to\infty$ we get\begin{equation}\nonumber\int_{\Omega}|u|^{p-1}u\,v=k\int_{\Omega}u\,v\end{equation}for every $v\in H^1_0(\Omega)$. Hence $u$ is constant, but this contradicts $u\in {\mathcal{M}}$.Now, if $u_n$ is a P.S. sequence for $f$ on ${\mathcal{U}}_{\alpha}$, there are sequences of real numbers $k_n$, $l_n$ such that\begin{equation}\label{ps1}\int_{\Omega}|u_n|^{p-1}u_n\,v-k_n\int_{\Omega}u_n\,v-l_n\int_{\Omega}\nabla u_n\,\nabla v=o(1)\,\|v\|_{H^1_0}.\end{equation}It is readily seen that $l_n$ is bounded away from zero, otherwise \eqref{ps1} is equivalent to \eqref{ps} (for some subsequence) and we still reach a contradiction. Then, we can divide both sides by $l_n$ and find thatthere are sequences $\{\lambda_n\}_n$, $\{\mu_n\}_n$, with $\mu_n$ bounded, such that\begin{equation}\nonumber\int_{\Omega}\nabla u_n\,\nabla v+\lambda_n\int_{\Omega}u_n\,v-\mu_n\int_{\Omega}|u_n|^{p-1}u_n\,v=o(1)\,\|v\|_{H^1_0}.\end{equation}Now, by reasoning as before one finds thatalso the sequence $\{\lambda_n\}_n$ is bounded, so that by the relation$$-\Delta u_n+\lambda_n u_n-\mu_n |u_n|^{p-1}u_n=o(1)\quad \mathrm{in}\,\, H^{-1}(\Omega)$$and by the compactness of the embedding $H^1_0(\Omega)\hookrightarrow L^{p+1}(\Omega)$, the P.S. condition holds for the functional $f\big |_{{\mathcal{U}}_{\alpha}}$.\end{proof}We can combine the previous lemmas to prove one of the main results stated in the introduction.\begin{proof}[Proof of Theorem \ref{thm:genus_2constr}]Lemma \ref{psball} allows to apply standard variational methods (see e.g.\cite[Thm. II.5.7]{St_2008}). We deduce that$M_{\alpha,\,k}$ is achievedat some critical point $u$ of $f\big |_{\mathcal{U}_{\alpha}}$.This amounts to say that $u$ satisfies \eqref{lagreq}for some real $\lambda$ and $\mu\neq 0$. We claim that there exists at least one$u\in f^{-1}(M_{\alpha,\,k})\cap \mathcal{U}_{\alpha}$ such that \eqref{lagreq} holds with $\mu>0$.Assume by contradiction that for \emph{every} critical point of $f\big |_{\mathcal{U}_{\alpha}}$ at level $M_{\alpha,k}$ it holds $\mu< 0$ in equation \eqref{lagreq}.Let us define the functional $T:\,H^1_0(\Omega)\to {\mathbb{R}}$ as$$T(u)=\frac{1}{2}\int_{\Omega}|\nabla u|^2.$$By denoting with $D$ the Fr\'{e}chet derivative and by $<\,,\,>$ the pairing between $H_0^1$ and its dual $H^{-1}$, our assumption can be restated as follows:\noindent if there are $u\in f^{-1}(M_{\alpha,\,k})\cap\mathcal{U}_{\alpha}$ and $\mu\neq 0$ such that\begin{equation}\label{lagreq1}\langle DT(u),\phi\rangle=\mu\langle Df(u),\phi\rangle\end{equation}for every $\phi\in H^1_0(\Omega)$ satisfying $\int_{\Omega}\phi u=0$ (that is for every $\phi$ tangentto ${\mathcal{M}}$ at $u$) then $\mu<0$.We stress that both $DT(u)$ and $Df(u)$ in the above equation are bounded away from zero, since there are no Dirichlet eigenfunctions in $\mathcal{U}_{\alpha}$ nor critical points of $f$ on ${\mathcal{M}}$.Hence, by denoting with $\nabla_{T{\mathcal{M}}}$ the gradient of a functional (in $H^1_0$) in the direction tangent to ${\mathcal{M}}$, if $u\in f^{-1}(M_{\alpha,\,k})\cap \mathcal{U}_{\alpha}$ then $\nabla_{T{\mathcal{M}}}T(u)$ and$\nabla_{T{\mathcal{M}}}f(u)$ \emph{are either opposite or not parallel}. Moreover, the angle between these (non vanishing) vectors is \emph{bounded away from zero}; otherwise, we would find sequences $u_n\in \mathcal{U}_{\alpha}$, $\mu_n>0$ such that\begin{equation}\label{noparal}(\nabla_{T{\mathcal{M}}}T(u_n),v)_{H^1_0}-\mu_n(\nabla_{T{\mathcal{M}}}f(u_n),v)_{H^1_0}=o(1)\|v\|_{H^1_0}\end{equation}for every $v\in H^1_0(\Omega)$; but since$$(\nabla_{T{\mathcal{M}}}T(u_n),v)_{H^1_0}=\int_{\Omega}\nabla u_n\,\nabla v-\lambda_n^T\int_{\Omega}u_n\, v\,,$$$$(\nabla_{T{\mathcal{M}}}f(u_n),v)_{H^1_0}=\int_{\Omega}|u_n|^{p-1}u_n\,v-\lambda_n^f\int_{\Omega}u_n\, v\,,$$for suitable bounded sequences $\lambda_n^T$, $\lambda_n^f$, this is equivalent to saying that $u_n$ is a P.S. sequence for$f\big |_{\mathcal{U}_{\alpha}}$, so that, by Lemma \ref{psball}, we would get a constrained critical point with $\mu>0$.Then, by choosing suitable linear combinations of the above tangential componentsone can define a bounded $\mathcal{C}^1$ map $u\mapsto v(u)\in H_0^1(\Omega)$, with $v(u)$ tangent to ${\mathcal{M}}$and satisfying the following property: there is $\delta>0$ such that\begin{equation}\label{diseqv}\int_{\Omega}\nabla u\,\nabla v(u)< -\delta\, ,\quad\quad \int_{\Omega}|u|^{p-1}u\,v(u)>\delta\,,\end{equation}for every $u\in f^{-1}(M_{\alpha,\,k})\cap \mathcal{U}_{\alpha}$. By continuity and possibly by decreasing $\delta$, inequalities \eqref{diseqv} extend to\begin{equation}\label{diseqv1}f^{-1}(M_{\alpha,\,k}-\bar\varepsilon, M_{\alpha,\,k}+\bar\varepsilon)\cap \big (\overline{\mathcal{B}}_{\alpha}\backslash \overline{\mathcal{B}}_{\alpha-\tau}\big )\end{equation}for small enough, positive $\bar\varepsilon$ and $\tau$. Finally, since there are no critical points of $f$ in${\mathcal{B}}_{\alpha}$ we can take that the \emph{second of \eqref{diseqv} holds on}\begin{equation}\label{diseqv2}f^{-1}(M_{\alpha,\,k}- \bar\varepsilon, M_{\alpha,\,k}+\bar\varepsilon)\cap \overline{\mathcal{B}}_{\alpha}.\end{equation} Let $\varphi$ be a $\mathcal{C}^1$ function on ${\mathbb{R}}$ such that:$$0\le\varphi\le 1, \quad\varphi\equiv 1\,\, \mathrm{in}\,\,(M_{\alpha,\,k}-\bar\varepsilon/2, M_{\alpha,\,k}+\bar\varepsilon/2),\quad \varphi\equiv 0\,\, \mathrm{in}\,\,{\mathbb{R}}\backslash (M_{\alpha,\,k}-\bar\varepsilon, M_{\alpha,\,k}+\bar\varepsilon),$$and define\begin{equation}\label{vectfield}e(u)=\varphi(f(u))\,v(u).\end{equation}Clearly, $e$ is a $\mathcal{C}^1$ vector field on ${\mathcal{M}}$ and is uniformly bounded, so that there exists a global solution $\Phi(u,t)$ of the initial value problem$$\partial_t\Phi(u,t)=e\big (\Phi(u,t)),\quad\quad \Phi(u,0)=0.$$By definition \eqref{vectfield} and by the first of \eqref{diseqv} (on \eqref{diseqv1}) we get$\Phi(u,t_0) \in \overline{\mathcal{B}}_{\alpha}$ for $t_0> 0$ and for any $u\in \overline{\mathcal{B}}_{\alpha}$; moreover, by the second inequality of \eqref{diseqv} (on \eqref{diseqv2}) there exists $\varepsilon\in (0,\bar\varepsilon)$ such that$$f(\Phi(u,t_0))>M_{\alpha,\,k}+\varepsilon$$ for every $u\in f^{-1}(M_{\alpha,\,k}-\varepsilon, +\infty)\cap \overline{\mathcal{B}}_{\alpha}$.Now, by \eqref{maxmin}, there is $A_{\varepsilon}\subset \overline{\mathcal{B}}_{\alpha}$ such that $\gamma(A_{\varepsilon})\ge k$ and$$\inf_{u\in A_{\varepsilon}} f(u)\ge M_{\alpha,\,k}-\varepsilon.$$Hence, $\gamma\big (\Phi(A_{\varepsilon},t_0) \big )\ge k$ and$$\inf_{u\in \Phi(A_{\varepsilon},t_0)} f(u)\ge M_{\alpha,\,k}+\varepsilon$$contradicting the definition of $M_{\alpha,\,k}$.\end{proof}\begin{remark}\label{rem1}If $\mu>0$, by testing \eqref{lagreq} with $u$ and by integration by parts we readily get$\lambda>-\alpha$. An alternative lower bound, independent of $\alpha$, could be obtainedby adapting arguments from \cite{MR968487,MR954951,MR991264} in order to provethat the Morse index of $u$ (as a solution of \eqref{lagreq}) is less or equal than $k$.Then Lemma \ref{lem:lambda_bdd_below} would provide $\lambda\geq-\lambda_{k}$.\end{remark}\begin{remark}\label{rem:MvsCNp}By the Gagliardo-Nirenberg inequality \eqref{sobest} we readily obtain that, for every $k\geq1$,\[M_{\alpha,k}\leq C_{N,p} \alpha^{N(p-1)/4}.\]Taking into account the previous remark, this agrees with Remark \ref{limGN}.\end{remark}We conclude this section with the following estimate.\begin{lemma}\label{lem:M3vsM1}Under the assumptions and notation of Theorem \ref{thm:genus_2constr},\[M_{\alpha,3} \leq 2^{-(p-1)/2} M_{\alpha,1}.\]\end{lemma}\begin{proof}Let $A\in \Sigma^{(3)}_{\alpha}$, according to Definition \ref{def:genus}. Noticethat the map\[A\ni u \mapsto \left(\int_\Omega |u|u , \int_\Omega |u|^p u\right)\in{\mathbb{R}}^2\]is continuous and equivariant. By the Borsuk-Ulam Theorem, we deduce the existence of$u_a\in A$ such that\[\int_\Omega |u^+_a|^2 = \int_\Omega |u^-_a|^2 = \frac12,\qquad\int_\Omega |u^+_a|^{p+1} = \int_\Omega |u^-_a|^{p+1} =\frac12 \int_\Omega |u_a|^{p+1},\]while\[\text{either }\int_\Omega |\nabla u^+_a|^2 \leq \frac{\alpha}{2}\qquad\text{or }\int_\Omega |\nabla u^-_a|^2 \leq \frac{\alpha}{2}.\]For concreteness let us assume that the first alternative holds;as a consequence, we obtain that $v:=\sqrt2 u_a^+$ belongs to$\overline{\mathcal{B}}_\alpha$. This yields\[M_{\alpha,1}\geq \int_\Omega |v|^{p+1} = 2^{(p+1)/2} \int_\Omega |u_a^+|^{p+1}= \frac{2^{(p+1)/2}}{2} \int_\Omega |u_a|^{p+1} \geq 2^{(p-1)/2}\inf_{u\in A} \int_\Omega |u|^{p+1},\]and since $A\in \Sigma^{(3)}_{\alpha}$ is arbitrary the proposition follows.\end{proof}\section{Min-max principles on the unit sphere in\texorpdfstring{$L^2$}{L\texttwosuperior}}\label{sec:1const}According to equation \eqref{Emu}, let ${\mathcal{M}}\subset H^1_0(\Omega)$ denote the unitsphere with respect to the $L^2$ norm and $\mathcal{E}_{\mu}$ the energy functionalassociated to \eqref{eq:main_prob_u}. In this section we are concerned with criticalpoints of $\mathcal{E}_{\mu}$ on ${\mathcal{M}}$ (which, in turn, correspond to solutions of ourstarting problem \eqref{eq:main_prob_U}).By the Gagliardo-Nirenberg inequality \eqref{sobest},setting $\|\nabla u\|^2_{L^2}=\alpha$, one obtains\begin{equation}\label{eq:boundonboundEmu}\frac12\,\alpha- \mu\frac{C_{N,p}}{p+1}\,\alpha^{N(p-1)/4}\leq \mathcal{E}_{\mu}(u)\le \frac12\alpha.\end{equation}In particular, $\mathcal{E}_{\mu}$ is bounded on any bounded subset of ${\mathcal{M}}$,and it is bounded from below (and coercive) on the entire ${\mathcal{M}}$ for {subcritical} $p<1+4/N$and for {critical} $p=1+4/N$ whenever $\mu< \frac{p+1}{2}C_{N,p}^{-1}$ .In these cases, one can easily show that $\mathcal{E}_{\mu}$ satifies the P.S. conditionand apply the classical {minimax principle for even functionals} on a closed symmetricsubmanifold (see e.g. \cite[Thm. II.5.7]{St_2008}).In the complementary case, when $p$ is either supercritical, i.e. $p>1+4/N$, orcritical and $\mu$ is large, then $\mathcal{E}_{\mu}$ is not bounded from below (see e.g.\eqref{minusinfty} below). In order to provide a minimax principle suitable for this case,we recall the Definition \ref{def:genus} of genus andthat of $\mathcal{B}_\alpha$ (see equation \eqref{eq:defBU}). Furthermore, we denote with $K_{c}$ the (closed and symmetric) set of critical points of${\mathcal{E}}_\mu$ at level $c$ contained in $\mathcal{B}_\alpha$. The following theorem is an adaptation of well known argumentsof previous critical point theorems relying on index theory.\begin{theorem}\label{infsupteo}Let $k\ge1$, $\alpha>\lambda_k(\Omega)$, $\mu>0$ and $\tau>0$ be fixed, and let $c_k$ be defined asin Theorem \ref{thm:genus_1constr}, equation \eqref{infsuplev}. If\begin{equation}\label{ass2} c_k < \hat c_k:= \inf_{\substack{A\in\Sigma^{(k)}_{\alpha}\\A\setminus\mathcal{B}_{\alpha-\tau}\neq\emptyset }}\sup_{A\setminus\mathcal{B}_{\alpha-\tau}}{\mathcal{E}}_\mu,\end{equation}then $K_{c_k}\neq\emptyset$, and it contains a critical point of Morse index less or equal to $k$.\end{theorem}\begin{remark}In case assumption \eqref{ass2} holds for $k,k+1,\dots,k+r$, and $c=c_k=...c_{k+r}$,then it is standard to extend Theorem \ref{infsupteo} to obtain\begin{equation}\label{indexK}\gamma(K_c)\ge r+1,\end{equation}so that $K_c$ contains infinitely many critical points.\end{remark}\begin{proof}[Proof of Theorem \ref{infsupteo}]For any $a\in{\mathbb{R}}$ we denote by ${\mathcal{M}}_a$ the sublevel set $\{\mathcal{E}_\mu<a\}$.First of all we notice that both $c_k$ and $\hat c_k$ are well defined and finite,by Lemma \ref{lemma:genusbigger} and equation \eqref{eq:boundonboundEmu}.Suppose now by contradiction that $K_{c_k}=\emptyset$. By a suitably modified version ofthe Deformation Lemma (recall that ${\mathcal{E}}_\mu$ satisfies the P.S. condition on ${\mathcal{M}}$),there exist $\delta>0$ and an equivariant homeomorphism$\eta$ such that$\eta(u)=u$ outside$\mathcal{B}_{\alpha}\cap {\mathcal{M}}_{c_k+2\delta}$ and\begin{equation}\label{lowlev}\eta({{\mathcal{M}}_{c_k+\delta}\cap \mathcal{B}_{\alpha-\tau}})\subset {\mathcal{M}}_{c_k-\delta}\cap\mathcal{B}_{\alpha}.\end{equation}By definition of $c_{k}$ there exists $A\in \Sigma^{(k)}_{\alpha}$ such that$A\subset {\mathcal{M}}_{c_k+\delta}$; it follows by assumption \eqref{ass2} (and by decreasing $\delta$ if necessary) that $A\subset {\mathcal{M}}_{c_k+\delta}\cap \mathcal{B}_{\alpha-\tau}$.Then, since $\eta$ is an odd homeomorphism, $\eta(A)\in \Sigma^{(k)}_{\alpha}$and, by definition, $\sup_{\eta(A)}{\mathcal{E}}_\mu \ge c_k$, in contradiction with\eqref{lowlev}. Finally, the estimate of the Morse index is a direct consequence of thedefinition of genus we deal with: see \cite{MR968487}, Proposition on page 1030, or the discussionat the end of Section 2 in \cite{MR991264}.\end{proof}We now provide a sufficient condition to guarantee the validity of assumption\eqref{ass2}.\begin{lemma}\label{lem:ckMak}Let $k\ge1$, $\alpha>\lambda_k(\Omega)$ and $\mu>0$ satisfy\begin{equation}\label{muboundef}0<\mu<\frac{p+1}{2}\,\frac{\alpha-\lambda_{k}(\Omega)}{M_{\alpha,k}-|\Omega|^{-\frac{p-1}{2}}}\end{equation}where $M_{\alpha,k}$ is defined in Theorem \ref{thm:genus_2constr}.Then, for $\tau>0$ sufficiently small, \eqref{ass2} holds true.\end{lemma}\begin{proof}We first estimate $c_k$ from above. To this aim, we construct a subset$\tilde A\in \Sigma^{(k)}_{\alpha-\tau}$ (for any $\tau$ sufficiently small)as\begin{equation}\label{Atilde}\tilde A= \left\{\sum_{i=1}^k x_i\varphi_i : x=(x_1,\dots,x_k)\in{\mathbb{S}}^{k-1}\right\},\end{equation}where, as usual $\varphi_i$ denotes the Dirichlet eigenfunction associated to$\lambda_i(\Omega)$. Indeed $\gamma(\tilde A)=k$ (it is homeomorphic to${\mathbb{S}}^{k-1}$), and $\max_{u\in\tilde A}\|u\|^2_{H^1_0}=\lambda_{k}(\Omega)<\alpha-\tau$ for$\tau$ small. Hence Holder inequality yields\begin{equation}\label{boundabove1}c_k \leq \sup_{\tilde A}\mathcal{E}_{\mu}\le \frac{1}{2}\lambda_{k}(\Omega)- \frac{\mu}{p+1}\,|\Omega|^{-\frac{p-1}{2}}.\end{equation}On the other hand, let $A\in\Sigma^{(k)}_{\alpha}$. Theorem \ref{thm:genus_2constr}implies\[\inf_{u\in A} \int_\Omega |u|^{p+1} \leq M_{\alpha,k}.\]If moreover $A\setminus\mathcal{B}_{\alpha-\tau}\neq\emptyset$ we infer\[\sup_{A\setminus\mathcal{B}_{\alpha-\tau}}\mathcal{E}_{\mu}\ge\frac{1}{2}(\alpha - \tau) - \frac{\mu}{p+1} M_{\alpha,k},\]and taking the infimum an analogous inequality holds true for $\hat c_k$. Comparingwith \eqref{boundabove1} the lemma follows.\end{proof}Exploiting the results above, we are ready to prove our main existence results.\begin{proof}[End of the proof of Theorem \ref{thm:genus_1constr}]By Theorem \ref{infsupteo} and Lemma \ref{lem:ckMak} the proof is completed by choosing\[\hat\mu_k:=\sup_{\alpha>\lambda_k(\Omega)} \frac{p+1}{2}\,\frac{\alpha-\lambda_{k}(\Omega)}{M_{\alpha,k}-|\Omega|^{-\frac{p-1}{2}}}.\qedhere\]\end{proof}\begin{proof}[Proof of Theorem \ref{thm:intro_GS}]We write the proof in terms of ${\mathcal{E}}_\mu$, the theorem following by the relations in\eqref{eq:main_prob_u}. Recall that, for every $u\in \overline{\mathcal{B}}_\alpha$, $\gamma\left(\{u,-u\}\right)=1$. We deduce that $c_1$ is actually a localminimum for ${\mathcal{E}}_\mu$, achieved by some $u$ which solves \eqref{eq:main_prob_u}(for a suitable $\lambda$), and it can be chosen positive by symmetry.Since\[\int_\Omega |\nabla u|^2 + \lambda u^2 - p\mu|u|^{p+1}\,dx=-(p-1)\int_\Omega \mu|u|^{p+1}\,dx<0,\]and $H^1_0(\Omega) = \spann\{u\}\oplus T_{u}\mathcal{M}$, we have that $u$has Morse index $1$. In a standard way, the minimalityproperty of $u$ implies also orbital stability of the associated solitary wave (see e.g.\cite{MR677997}). Turning to the estimates for $\hat\mu_1 = \hat\rho_1^{(p-1)/2}$,we can deduce it using Lemma \ref{lem:ckMak} and Remark \ref{rem:MvsCNp}, which yield\[\hat\mu_1\left(\Omega,p\right):=\sup_{\alpha>\lambda_1(\Omega)} \frac{p+1}{2}\,\frac{\alpha-\lambda_{1}(\Omega)}{C_{N,p} \alpha^{\frac{N(p-1)}{4}}-|\Omega|^{-\frac{p-1}{2}}}\geq \frac{p+1}{2C_{N,p}}\,\sup_{\alpha>\lambda_1(\Omega)}\frac{\alpha-\lambda_{1}(\Omega)}{\alpha^{\beta}},\]where $\beta:=N(p-1)/4$. Now, if $\beta\leq1$ we obtain the desired bound for thesubcritical and critical cases. On the other hand, when $\beta>1$, elementarycalculations show that\[\hat\mu_1\left(\Omega,p\right)\geq \frac{p+1}{2C_{N,p}}\,\frac{(\beta-1)^{(\beta-1)}}{\beta^\beta}\, \lambda_1(\Omega)^{-(\beta-1)},\]and finally\[\hat\rho_1\left(\Omega,p\right)\geq \underbrace{\left[\frac{p+1}{2C_{N,p}}\,\frac{(\beta-1)^{(\beta-1)}}{\beta^\beta}\right]^{\frac{2}{p-1}}}_{D_{N,p}}\,\lambda_1(\Omega)^{\frac{2}{p-1}-\frac{N}{2}}.\qedhere\]\end{proof}\begin{proof}[Proof of Proposition \ref{thm:intro_3>1}]As usual, by \eqref{eq:main_prob_u}, we have to prove that\[\hat\mu_3\left(\Omega,p\right)\geq2^{(p-1)/2} D_{N,p}\lambda_3(\Omega)^{\frac{2}{p-1}-\frac{N}{2}}.\]By Lemmas \ref{lem:ckMak}, \ref{lem:M3vsM1}, and Remark \ref{rem:MvsCNp} we obtain\[\begin{split}\hat\mu_3 &= \sup_{\alpha>\lambda_3(\Omega)} \frac{p+1}{2}\,\frac{\alpha-\lambda_{3}(\Omega)}{M_{\alpha,3}-|\Omega|^{-\frac{p-1}{2}}} \geq\sup_{\alpha>\lambda_3(\Omega)} \frac{p+1}{2}\,\frac{\alpha-\lambda_{3}(\Omega)}{2^{-(p-1)/2}M_{\alpha,1}-|\Omega|^{-\frac{p-1}{2}}}\\&\geq 2^{(p-1)/2}\sup_{\alpha>\lambda_3(\Omega)} \frac{p+1}{2}\,\frac{\alpha-\lambda_{3}(\Omega)}{C_{N,p}\alpha^\beta-2^{(p-1)/2}|\Omega|^{-\frac{p-1}{2}}},\end{split}\]where $\beta:=N(p-1)/4$, and the desired result follows by arguing as in the proof ofTheorem \ref{thm:intro_GS}.\end{proof}To conclude this section we prove that in the supercritical case, if $\mu$ is not toolarge, in addition to $(c_k)_k$ there is a further sequence of critical levels$(\bar c_k)$ of $\mathcal{E}_{\mu}$ constrained to $\mathcal{M}$. For concreteness,let us first consider the case $k=1$: since in such case $c_1$ is a local minimum of${\mathcal{E}}_\mu$ in $\mathcal{M}$, and ${\mathcal{E}}_\mu$ is unbounded from below in $\mathcal{M}$,the critical level $\bar c_1$ is of mountain pass type.\begin{proposition}\label{mpcritlev}Let $p>1+4/N$, $\mu<\hat\mu_1$, and $u_1$ denote the local minimum point of ${\mathcal{E}}_\mu$in $\mathcal{M}$, according to Theorems \ref{infsupteo} and \ref{thm:intro_GS}. The value\[\bar c_1 : =\inf_{\gamma\in \Gamma}\sup_{[0,1]}\mathcal{E}_{\mu}(\gamma(s)),\quad\text{where }\Gamma:=\left\{\gamma\in C([0,1];{\mathcal{M}}) : \gamma(0)=u_1,\,\gamma(1)<c_1-1\right\},\]is a critical level for ${\mathcal{E}}_\mu$ in $\mathcal{M}$.\end{proposition}\begin{proof}Notice that, if $p>1+4/N$, then $\mathcal{E}_{\mu}\to -\infty$ along some sequence in${\mathcal{M}}$. Indeed, by defining\begin{equation}\label{concfun}w_n(x): = \eta(x)Z_{N,p}\big ((x-x_0)/a_n\big )\quad \text{and}\quad\tilde{w_n}:=\frac{w_n}{\| w_n\|^2_{L^2(\Omega)}}\,\in\, {\mathcal{M}},\end{equation}where $a_n\to 0^+$, $x_0\in \Omega$ and $\eta\in \mathcal{C}_0^{\infty}(\Omega)$, $\eta(x_0)=1$, we obtain\begin{equation}\label{minusinfty}\alpha_n :=\|\nabla \tilde w_n\|^2_{L^2(\Omega)}\to +\infty,\qquad\frac{\int_{\Omega} |\tilde w_n|^{p+1} \,dx}{\alpha_n^{N(p-1)/4}}\to C_{N,p},\qquad\mathcal{E}_{\mu}(\tilde w_n)\to -\infty\end{equation}for $n\to +\infty$. Since $u_1$ is a local minimum,the functional $\mathcal{E}_{\mu}$ has a mountain pass structure on ${\mathcal{M}}$;by recalling that $\mathcal{E}_{\mu}$ satisfies the P.S. condition theproposition follows.\end{proof}\begin{remark}\label{rem:further_crit_lev}One can generalize Proposition \ref{mpcritlev} by constructing critical points via a saddle-point theorem in the following way: let us pick $k$ points $x_1, x_2,...,x_k$ in $\Omega$ and consider the corresponding function $\tilde w_i$; we may assume that$\mathrm{supp}\,\tilde w_i\cap \mathrm{supp}\,\tilde w_j=\emptyset$ for $i\neq j$, so that these functions are orthogonal. Let us now define the subspace$V_k=\mathrm{span}\{\varphi_1,\dots,\varphi_k;\tilde w_1,...,\tilde w_k\}$; note that dim $V_k=2k$. Let $R$ be an operator (in $L^2(\Omega)$) such that $R=I$ on $V_k^{\perp}$, $Ru_i=\tilde w_i$, $i=1,2,..,k$. Possibly after permutations, we can choose $R$ such that $R\big |_{V_k}\in SO(2k)$ (actually, there are infinitely many different choices of $R$). Now, since $SO(2k)$ is (arcwise) connected, there is a continuous path$\tilde{\gamma}:\,[0,1]\rightarrow SO(2k)$ such that $\gamma(0)=I$, $\gamma(1)=R\big |_{V_k}$.Then, we can define the following map\[\gamma:\, [0,1] \times S^{k-1}\rightarrow {\mathcal{M}},\quad\quad \gamma(s;t_1,....,t_k)=\sum_{i=1}^{k}t_i\tilde{\gamma}(s)u_i,\quad\]where $\sum_{i=1}^{k}t_i^2=1$. It is clear that $\gamma$ is continuous; moreover,\[\gamma(0;t_1,....,t_k)\in \mathrm{span }\{\varphi_1,\dots,\varphi_k\}\cap {\mathcal{M}}\quad \mathrm{and}\quad\gamma(1;t_1,....,t_k)\in \mathrm{span }\{\tilde w_1,\dots,\tilde w_k\}\cap {\mathcal{M}}.\]Then, by denoting with $\Gamma_k$ the set of the above paths, if $\mu$ is sufficientlysmall we obtain the critical levels\[\bar c_k : =\inf_{\gamma\in \Gamma_k}\sup_{[0,1]\times S^{k-1}}\mathcal{E}_{\mu}(\gamma(s;t_1,....,t_k)).\]\end{remark}\section{Results in symmetric domains}\label{sec:symm}This section is devoted to the proof of Theorem \ref{pro:symm}, therefore we assume $1+4/N \leq p < 2^*-1$.We perform the proof in the case of $\Omega=B$, but it will be clear that the main assumption on $\Omega$ is the following:\begin{itemize} \item[\textbf{(T)}] there is a tiling of $\Omega$, made by $h$ copies of a subdomain $D$, in such a way that from any solution $U_D$ of \eqref{eq:main_prob_U} on $D$ one can construct, using reflections, a solution $U_\Omega$ of \eqref{eq:main_prob_U} on $\Omega$.\end{itemize}Then $U_\Omega$ has $h$ times the mass of $U_D$, and recalling Theorem \ref{thm:intro_GS} we deduce that \eqref{eq:main_prob_U} on $\Omega$ is solvable for any$\rho< h \cdot D_{N,p} \lambda_1(D)^{\frac{2}{p-1}-\frac{N}{2}}$. At this point, for a sequence $(D_k,h_k)_k$ of tilings satisfying \textbf{(T)}, weobtain the solvability of \eqref{eq:main_prob_U} on $\Omega$ whenever\[\rho< h_k \cdot D_{N,p} \lambda_1(D_k)^{\frac{2}{p-1}-\frac{N}{2}},\]and if we can show that\begin{equation}\label{eq:finaltarget}\frac{ h_k }{ \lambda_1(D_k)^{\frac{N}{2}-\frac{2}{p-1}}} \to +\infty\qquad\text{as }k\to+\infty,\end{equation}we deduce the solvability of \eqref{eq:main_prob_U} on $\Omega$ for every mass. Having this scheme in mind, it is easy to prove analogous resultson rectangles and also in other kind of domains.Then let $B\subset{\mathbb{R}}^N$ be the ball (w.l.o.g. of radius one), and let\[D_k:=\left\{(r\cos\theta, r\sin\theta,x_3,\dots,x_N)\in B: - \frac{\pi}{k} < \theta < \frac{\pi}{k}\right\}\]Then $D_k$ satisfies \textbf{(T)}, with $h_k=k$. In order to estimate $\lambda_1(D_k)$ we observe that, by elementary trigonometry,\[B'_k = B_{\frac{\sin(\pi/k)}{\sin(\pi/k)+1}}\left(\frac{1}{\sin(\pi/k)+1},0,0,\dots,0\right) \subset D_k,\]and therefore\[\lambda_1(D_k) \le \lambda_1(B'_k) \le C k^2,\]for some dimensional constant $C=C(N)$ and $k$ large. Then\[\frac{ h_k }{ \lambda_1(D_k)^{\frac{N}{2}-\frac{2}{p-1}}}\ge C \frac{k}{k^{{N}-\frac{4}{p-1}}} = C k^{1-{N}+\frac{4}{p-1}} = C k^{\frac{N-1}{p-1}\left[1+\frac{4}{N-1} - p\right]},\]and finally \eqref{eq:finaltarget} holds true whenever $p< 1+\frac{4}{N-1}$, thus completing the proof of Theorem \ref{pro:symm}.\small\subsection*{Acknowledgments}We would like to thank Jacopo Bellazzini, who pointed out that the results in\cite{MR3318740}, in the supercritical case, can be read in terms of a local minimization. Wewould also like to thank Benedetta Noris, who read a preliminary version of this manuscript.This work is partially supported by the PRIN-2012-74FYK7 Grant:``Variational and perturbative aspects of nonlinear differential problems'',by the ERC Advanced Grant 2013 n. 339958:``Complex Patterns for Strongly Interacting Dynamical Systems - COMPAT'',and by the INDAM-GNAMPA group.

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