\documentclass[reqno]{amsart} \usepackage{amssymb} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2003(2003), No. 81, pp. 1--14.\newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \thanks{\copyright 2003 Southwest Texas State University.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE--2003/81\hfil Concentration and dynamic system] {Concentration and dynamic system of solutions for semilinear elliptic equations} \author[Tsung-fang Wu\hfil EJDE--2003/81\hfilneg] {Tsung-fang Wu} \address{Tsung-fang Wu \newline Center for General Education\\ Southern Taiwan University of Technology, Taiwan} \email{tfwu@mail.stut.edu.tw} \date{} \thanks{Submitted June 23, 2003. Published August 7, 2003.} \thanks{Partially supported by the National Science Council of China} \subjclass[2000]{35J20, 35J25, 35J60} \keywords{Palais-Smale, concentration, dynamic system, multiple solutions} \begin{abstract} In this article, we use the concentration of solutions of the semilinear elliptic equations in axially symmetric bounded domains to prove that the equation has three positive solutions. One solution is $y$-symmetric and the other are non-axially symmetric. We also study the dynamic system of these solutions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} Consider the semilinear elliptic equation $$\begin{gathered} -\Delta u+u=|u|^{p-2}u \quad\text{in }\Omega , \\ u\in H_{0}^{1}(\Omega ), \end{gathered} \label{E1}$$ where $N\geq 2$, $2^{*}=\frac{2N}{N-2}$ for $N\geq 3$ and $2^{*}=\infty$ for $N=2$, $20$. Moreover, $\mathbf{A}_{-t,t}^{r}$ is convex in $x$ and in $y$. Thus, by Gidas-Ni-Nirenberg \cite{GNN}, every positive solution of equation (\ref{E1}) in $\mathbf{A}_{-t,t}^{r}$ for each $t>0$ is radially symmetric in $x$ and axially symmetric in $y$. Actually, Dancer \cite{Da} proved that the positive solution of equation (\ref{E1}) in $\mathbf{A}_{-t,t}^{r}$ for each $t>0$ in $\mathbb{R}^{2}$ is unique. However, the axially symmetry and uniqueness of positive solution generally fails if $\Omega$ is not convex in the $y$-direction. First, we consider a perturbation of the finite strip $\mathbf{A}_{-t,t}^{r}$, that is dumbbell type domain $D=B^{N}((0;-t) ,r_{0}) \cup \mathbf{A}_{-t,t}^{r}\cup B^{N}((0;t) ,r_{0}) \quad \quad\text{for } B^{N-1}(0;r) \subset B^{N-1}(0;r_{0}) .$ Then the dumbbell domain $D$ is symmetric in $y-$axis, but not convex in $y$-direction. Moreover, the Dancer \cite{Da} and Byeon \cite{By1}, \cite{By2} proved that the equation $(\ref{E1})$ in $D$ has at least three positive solutions, for $B^{N-1}(0;r)$ is sufficiently close to a point $x_{0}$ in $\mathbb{R}^{N-1}$. And Chen-Ni-Zhou \cite{CNZ} use computational showed that the equation $(\ref{E1})$ in some dumbbell-type domains has multiple positive solutions and describe the concentration of these solutions. The main purpose of this paper is using the Palais-Smale theory to present another perturbation. Let $\omega$ be a $y$-symmetric bounded set such that $\mathbf{A}^{r}\backslash \overline{\omega }\subsetneqq \mathbf{A}^{r}$ is a domain in $\mathbb{R}^{N}$ for some $t>0$, consider the finite strip with holes $\Theta _{t}=\mathbf{A}_{-t,t}^{r}\backslash \overline{\omega }.$ Then there exists a $t'>0$ such that $\Theta _{t}$ is also symmetric in $y-$axis, but not convex in $y-$direction for each $t>t'$. We prove that there exists a $t_{0}>0$ such that for $t\geq t_{0}$, the equation $(\ref{E1})$ in $\Theta _{t}$ has three positive solutions which one is $y$-symmetric and the other are non-axially symmetric. Moreover, we describe the concentration and dynamic system of these solutions. Although, Wang-Wu \cite{WW} used the symmetry of positive solutions showed the same multiple results in a finite strip with hole $\mathbf{A}_{-t,t}^{r}\backslash B^{N}(0;r')$ for $t$ sufficiently large. However, they have not describe the concentration and dynamic system of solutions. This article is organized as follow. In section 2, we describe various preliminaries. In section 3, we describe various compactness results. In section 4, we describe some properties of the large domains in $\mathbf{A}^{r}$. In section 5 and section 6, we present the concentration and dynamic system of the solutions. \section{Preliminary} In this article, we focus on the problems on two Hilbert spaces: the whole Sobolev space $H_{0}^{1}(\Omega )$ and its closed linear subspace $H_{s}(\Omega )$ defined as follows: Let $z=(x,y)\in \mathbb{R}^{N-1}\times \mathbb{R}$ and $\Omega$ be a domain in $\mathbb{R}^{N}$. \begin{definition} \label{p1} \rm \begin{itemize} \item[(i)] $\Omega$ is $y$-symmetric provided $\,z=(x,y)\in \Omega$ if and only if $(x,-y)\in \Omega$; \item[(ii)] Let $\Omega$ be a $y$-symmetric domain in $\mathbb{R}^{N}$. A function $u:\Omega \to \mathbb{R}$ is $y$-symmetric (axially symmetric) if $u(x,y)=u(x,-y)$ for $(x,y)\in \Omega$. \end{itemize} \end{definition} In this article, we let $\Omega$ be a $y$-symmetric domain in $\mathbb{R}^{N}$ and $H_{s}(\Omega )$ the $H^{1}$-closure of the space $\{u\in C_{0}^{\infty }(\Omega ) : u \quad\text{is$y$-symmetric}\}$ and let $X(\Omega)$ be either the whole space $H_{0}^{1}(\Omega)$ or the $y$-symmetric Sobolev space $H_{s}(\Omega)$. Then $H_{s}(\Omega )$ is a closed linear subspace of $H_{0}^{1}(\Omega )$. Let $H_{s}^{-1}(\Omega)$ be the dual space of $H_{s}(\Omega )$. We define the Palais-Smale (simply by (PS)) sequences, (PS)-values and (PS)-conditions in $X(\Omega)$ for $J$ as follows. \begin{definition} \label{p2} \rm \begin{itemize} \item[(i)] For $\beta \in \mathbb{R}\mathbf{,}$ a sequence $\{ u_{n}\}$ is a (PS)$_{\beta }$-sequence in $X(\Omega)$ for $J$ if $% J(u_{n})=\beta +o(1)\;$and$\;J'(u_{n})=o(1)\;$strongly in $% X^{-1}(\Omega)$ as $n\to \infty$ \item[(ii)] $\beta \in \mathbb{R}$ is a (PS)-value in $X(\Omega)$ for $J$ if there is a (PS)$_{\beta }$-sequence in $X(\Omega)$ for $J$ \item[(iii)] $J$ satisfies the (PS)$_{\beta }-$condition in $X(\Omega)$ if every (PS)$_{\beta }$-sequence in $X(\Omega)$ for $J$ contains a convergent subsequence. \end{itemize} \end{definition} Now, we consider the Nehari minimization problem $\alpha _{X}(\Omega )=\inf_{u\in \mathbf{M}(\Omega )}J(u),$ where $\mathbf{M}(\Omega )=\left\{ u\in X(\Omega) \backslash \{0\}:a(u)=b(u)\right\}$. Note that $\mathbf{M}(\Omega )$ contains every nonzero solution of equation (\ref{E1}) in $\Omega$, $\alpha _{X}(\Omega )>0$, and if $u_{0}\in \mathbf{M}(\Omega )$ achieves $\alpha _{X}(\Omega )$, then $u_{0}$ is a positive (or negative) solution of equation (\ref{E1}) in $\Omega$ (see Wang-Wu \cite{WW} or Willem \cite{Wi}). We have the following useful lemma, whose proof can be found in Wang-Wu \cite[Lemma 7]{WW}. \begin{lemma} \label{p7} Let $\{u_{n}\}$ be in $X(\Omega)$. Then $\{u_{n}\}$ is a (PS)$_{\alpha _{X}(\Omega )}$-sequence in $X(\Omega)$ for $J$ if and only if $J(u_{n}) =\alpha _{X}(\Omega )+\mathrm{o}% (1)$ and $a(u_{n}) =b(u_{n}) +\mathrm{o}% (1)$. In particular, every minimizing sequence $\{u_{n}\}$ in $\mathbf{M}(\Omega )$ for $\alpha _{X}(\Omega )$ is a \newline (PS)$_{\alpha _{X}(\Omega )}$-sequence in $X(\Omega )$ for $J$. \end{lemma} We denote $\alpha _{X}(\Omega )$ by $\alpha (\Omega )$ for $X(\Omega) =H_{0}^{1}(\Omega)$. We denote $\alpha _{X}(\Omega )$ by $\alpha _{s}(\Omega )$ for $X(\Omega) =H_{s}(\Omega)$. We denote $\mathbf{M}(\Omega )$ by $\mathbf{M}_{0}(\Omega )$ for $X(\Omega) =H_{0}^{1}(\Omega)$. We denote $\mathbf{M}(\Omega )$ by $\mathbf{M}_{s}(\Omega )$ for $X(\Omega) =H_{s}(\Omega)$. \begin{remark} \rm By the Principle of symmetric criticality (see Palais \cite{P}), we have a (PS)$_{\beta }$-sequence in $X(\Omega)$ for $J$ is a (PS)$% _{\beta }$-sequence in $H_{0}^{1}(\Omega)$ for $J$. \end{remark} \section{Palais-Smale Conditions} In this section, we present several (PS)$_{\alpha _{X}(\Omega) }-$conditions in $X(\Omega )$ for $J$ which are used to prove our main results in section 4 and section 5. Since for each (PS)$_{\alpha _{X}(\Omega) }$-sequence $\{ u_{n}\}$ in $X(\Omega )$ for $J$, there exists a subsequence $\{ u_{n}\}$ and $u$ in $% X(\Omega )$ such that $u_{n}\rightharpoonup u$ weakly in $X(\Omega )$. Then $u$ is a solution of equation (\ref{E1}) in $\Omega$. Moreover, we have the following result, whose proof can be found in Bahri-Lions \cite{BL} and in Wang-Wu \cite{WW}. \begin{lemma} \label{c4} For each (PS)$_{\alpha _{X}(\Omega) }$-sequence $\{u_{n}\}$ in $X(\Omega)$ for $J$, there exists a subsequence $\{u_{n}\}$and a nonzero $u$ in $X(\Omega )$ such that $u_{n}\rightharpoonup u$ weakly in $X(\Omega)$ if and only if the (PS)$_{\alpha _{X}(\Omega) }-$condition holds in $X(\Omega )$ for $J$. \end{lemma} Let $\Omega$ be any unbounded domain and $\xi \in C^{\infty}([0,\infty ))$ such that $0\leq \xi \leq 1$ and $\xi (t)=\begin{cases} 0, & \text{for }t\in [0,1] \\ 1, & \text{for }t\in [2,\infty ). \end{cases}$ Let $$\xi _{n}(z)=\xi (\frac{2|z|}{n}). \label{11}$$ Then we have the following results. \begin{proposition} \label{c1} The equation $(\ref{E1})$ in $\Omega$ does not admit any solution $u_{0}$ such that $J(u_{0}) =\alpha _{X}(\Omega )$ if and only if for each (PS)$_{\alpha _{X}(\Omega )}$-sequence $\{u_n\}$ in $X(\Omega)$ for $J$, there exists a subsequence $\{u_n\}$ such that $\{ \xi_{n}u_{n}\}$ is also a (PS)$_{\alpha _{X}(\Omega )}$-sequence in $X(\Omega)$ for $J$. \end{proposition} \begin{proof} Let $\{u_n\}$ be a (PS)$_{\alpha _{X}(\Omega )}$-sequence in $X(\Omega)$ for $J$. Then there exist a subsequence $\{u_n\}$ and $u_{0}\in X(\Omega)$ such that $u_{n}\rightharpoonup u_{0}$ weakly in $X(\Omega)$. Since the equation $(\ref{E1})$ in $\Omega$ does not admit any solution $u_{0}$ such that $J(u_{0}) =\alpha _{X}(\Omega )$, by Lemma \ref {c4}, we have $u_{0}=0$. Let $v_{n}=\xi _{n}u_{n}$. First, we need to show $$a(u_{n}-v_{n}) =o(1) . \label{11-2}$$ Note that $a(u_{n}-v_{n}) =a(u_{n}) +a(v_{n}) -2\left\langle u_{n},v_{n}\right\rangle _{H^{1}}.$ Thus, it suffices to show that $\left\langle u_{n},v_{n}\right\rangle _{H^{1}}=a(u_{n}) +o(1) =a(v_{n})+o(1)$. Since \begin{align*} \left\langle u_{n},v_{n}\right\rangle _{H^{1}} &=\int_{\Omega }\nabla u_{n}\nabla v_{n}+u_{n}v_{n} \\ &=\int_{\Omega }\xi _{n}\left[ | \nabla u_{n}| ^{2}+u_{n}^{2}\right] +\int_{\Omega }u_{n}\nabla u_{n}\nabla \xi _{n}. \end{align*} Note that $|\nabla \xi _{n}| \leq \frac{c}{n}$ and $\{u_n\}$ is a (PS)$_{\alpha _{X}(\Omega ) }$-sequence in $X(\Omega)$ for $J$, so $$\int_{\Omega }\xi _{n}^{q}u_{n}\nabla u_{n}\nabla \xi _{n}=o(1) \quad \text{for }q>0. \label{12}$$ Hence, $$\left\langle u_{n},v_{n}\right\rangle _{H^{1}}=\int_{\Omega }\xi _{n}\left[ | \nabla u_{n}| ^{2}+u_{n}^{2}\right] +o(1) . \label{13}$$ Similarly, we have $$a(v_{n}) =\int_{\Omega }\xi _{n}^{2}\left[ | \nabla u_{n}| ^{2}+u_{n}^{2}\right] +o(1) . \label{14}$$ For $r\geq 1$. Since $\{\xi _{n}^{r}u_{n}\}$ is bounded in $X(\Omega )$, we have \begin{align*} o(1) &=\left\langle J'(u_{n}),\xi _{n}^{r}u_{n}\right\rangle \\ &=\int_{\Omega }(\xi _{n}^{r}|\nabla u_{n}|^{2}+r\xi _{n}^{r-1}u_{n}\nabla \xi _{n}\nabla u_{n}+\xi _{n}^{r}u_{n}^{2})-\int_{\Omega }\xi _{n}^{r}|u_{n}|^{p}. \end{align*} By $(\ref{12})$, we conclude that $$\int_{\Omega }\xi _{n}^{r}(|\nabla u_{n}|^{2}+u_{n}^{2})=\int_{\Omega }\xi _{n}^{r}|u_{n}|^{p}+o(1). \label{15}$$ Since $u_{n}\rightharpoonup 0$ weakly in $H_{0}^{1}(\Omega )$, there exists a subsequence $\{u_n\}$ such that $u_{n}\to 0$ strongly in $L_{loc}^{p}(\Omega)$, or there exists a subsequence $\{u_n\}$ such that $\int_{Q(n) }|u_n| ^{p}=o(1),$ where $Q(n) =\Omega \cap B^{N}(0;n)$. Clearly, $$\int_{\Omega }\xi _{n}^{r}|u_{n}|^{p}=\int_{\Omega }|u_{n}|^{p}+o(1). \label{15-1}$$ By $(\ref{13}) ,(\ref{14}), (\ref{15})$ and $(\ref{15-1})$, we have $\left\langle u_{n},v_{n}\right\rangle _{H^{1}}=a(u_{n}) +o(1) =a(v_{n}) +o(1) .$ Moreover, by the compact imbedding theorem, we obtain $$b(v_{n}) =b(u_{n}) +o(1) . \label{15-2}$$ Since $a(u_{n}) =b(u_{n}) +o(1)$. Thus, from $(\ref{11-2})$ and $(\ref{15-2})$, we obtain \begin{gather*} a(v_{n}) =b(v_{n}) +(1)\,,\ J(v_{n}) =\alpha _{X}(\Omega )+o(1) \,. \end{gather*} By Lemma \ref{p7}, we can conclude that $\left\{ \xi _{n}u_{n}\right\}$ is a (PS)$_{\alpha _{X}(\Omega )}$-sequence in $X(\Omega)$ for $% J.$ Conversely, assume that the equation $(\ref{E1})$ in $% \Omega$ admits a solution $u_{0}$ such that $J(u_{0}) =\alpha _{X}(\Omega )$. We may assume that $u_{0}$ is a positive solution. Let $% u_{n}=u_{0}$ for each $n\in \mathbb{N}$, then $\{u_n\}$ is a (PS)% $_{\alpha _{X}(\Omega )}$-sequence in $X(\Omega)$ for $J$. By hypothesis, we have $\left\{ \xi _{n}u_{0}\right\}$ is also a (PS)$_{\alpha _{X}(\Omega )}$-sequence in $X(\Omega)$ for $J$. We obtain $\int_{\Omega }| \xi _{n}u_{0}| ^{p}=\frac{2p}{p-2}\alpha _{X}(\Omega )+o(1) .$ Thus, there exist $n_{0}$ and $d>0$ such that $$\int_{\Omega }| \xi _{n}u_{0}| ^{p}>d\quad\text{for each }n\geq n_{0}. \label{15-3}$$ However, $u_{0}\in L^{p}(\Omega)$. Hence $\int_{\Omega }| \xi _{n}u_{0}| ^{p}\leq \int_{\left[ B^{N}(0;\frac{n}{2}) \right] ^{c}}| u_{0}| ^{p}=o(1) \quad\text{as }n\to \infty ,$ this contradicts to $(\ref{15-3})$. \end{proof} \begin{proposition} \label{c2} $J$ does not satisfy the (PS)$_{\alpha _{X}(\Omega )}-$condition in $X(\Omega)$ for $J$ if and only if there exists a (PS)$% _{\alpha _{X}(\Omega )}$-sequence $\{u_{n}\}$ in $X(\Omega)$ for $J$ such that $\left\{ \xi _{n}u_{n}\right\}$ is also a (PS)$_{\alpha _{X}(\Omega )}$-sequence in $X(\Omega)$ for $J$. \end{proposition} The proof of this proposition is similar to the proof of Proposition \ref{c1} and therefore, it is omitted. Let $\Omega _{1}\subsetneqq \Omega _{2}$, clearly $\alpha _{X}(\Omega _{1}) \geq \alpha _{X}(\Omega _{2})$. Then we have the following useful results. \begin{lemma}\label{c5} Let $\Omega _{1}\subsetneqq \Omega _{2}$ and $J:X(\Omega _{2})\to \mathbb{R}$ be the energy functional. Suppose that $\alpha _{X}(\Omega _{1}) =\alpha _{X}(\Omega _{2})$. Then \begin{itemize} \item[(i)] The equation $(\ref{E1})$ in $\Omega _{1}$ does not admit any solution $u_{0}$ such that $J(u_{0}) =\alpha _{X}(\Omega _{1})$ \item[(ii)] $J$ does not satisfy the (PS)$_{\alpha _{X}(\Omega _{2})}-$condition. \end{itemize} \end{lemma} The proof of this lemma can be found in Wang-Wu \cite[Lemma 13]{WW}. By the Rellich compact theorem, $J$ satisfies the (PS)$_{\alpha _{X}(\Omega )}-$condition in $X(\Omega )$ if $\Omega$ is a bounded domain. \begin{lemma} \label{c8} Let $\Omega$ be a bounded domain in $\mathbb{R}^{N}$. Then the (PS)$% _{\alpha _{X}(\Omega )}-$condition holds in $X(\Omega )$ for $J$. Furthermore, the equation $(\ref{E1})$ in $\Omega$ has a positive solution $u_{0}$ such that $J(u_{0}) =\alpha _{X}(\Omega )$. \end{lemma} \section{Large Domains in $\mathbf{A}^r$} \begin{definition} \label{e1} \rm A domain $\Omega$ in $\mathbf{A}^{r}$ is large if for any $m>0$, there exist $s0$, $\Omega _{1}$ and $\Omega _{2}$ are large domains in $\mathbf{A}^{r}$ such that $\Omega \backslash \overline{\mathbf{A}% _{-t_{0},t_{0}}^{r}}=\Omega _{1}\cup \Omega _{2}$. Let $\left\{ u_{n}^{1}\right\}$ be a (PS)$_{\alpha (\Omega _{1}) }$-sequence in $H_{0}^{1}(\Omega _{1})$ for $J$ and let $u_{n}^{2}(x,y) =u_{n}^{1}(x,-y)$. Clearly, $\left\{ u_{n}^{2}\right\}$ is a (PS)$_{\alpha (\Omega _{2}) }$-sequence in $H_{0}^{1}(\Omega _{2})$ for $J$. Take $v_{n}=u_{n}^{1}+u_{n}^{2}$, then $v_{n}\in H_{s}(\Omega)$, $a(v_{n}) =b(v_{n}) +o(1)$ and $J(v_{n}) =\alpha (\Omega _{1}) +\alpha (\Omega _{2}) +o(1).$ Moreover, there exists $s_{n}>0$ such that $s_{n}v_{n}\in \mathbf{M}_{s}(\Omega)$ and $J(s_{s}v_{n}) =\alpha (\Omega _{1}) +\alpha (\Omega _{2}) +o(1).$ From Lemma \ref{e4} and the definition of Nehari minimization problem, we can conclude $\alpha _{s}(\Omega) \leq 2\alpha (\mathbf{A}^{r})$. \end{proof} Then we have the following symmetric compactness. \begin{proposition}\label{e7} Suppose that $\Omega$ is a $y$-symmetric large domain in $\mathbf{A}^{r}$. Then $J$ satisfies the (PS)$_{\alpha _{s}(\Omega )}-$condition in $% H_{s}(\Omega)$ if and only if $\alpha _{s}(\Omega) <2\alpha (\mathbf{A}^{r})$. \end{proposition} \begin{proof} Suppose that $J$ satisfies the (PS)$_{\alpha _{s}(\Omega )}-$condition in $% H_{s}(\Omega )$. By Lemma \ref{e6}, we have $\alpha _{s}(\Omega) \leq 2\alpha (\mathbf{A}^{r})$. Suppose that $\alpha_{s}(\Omega) =2\alpha (\mathbf{A}^{r})$. By the definition of domain in $\mathbb{R}^{N}$, we may take a domain $\tilde{\Omega}=\Omega \backslash \overline{B^{N}(0;\tilde{r}) }$ for some $\tilde{r}>0$ such that $\tilde{\Omega}\subsetneqq \Omega$ and $\tilde{\Omega}$ is a proper $y$-symmetric large domain in $\mathbf{A}^{r}$. By Lemma \ref{c5}, we have $2\alpha (\mathbf{A}^{r}) =\alpha _{s}(\Omega) <\alpha _{s}(\tilde{\Omega})$. This contradicts to Lemma \ref{e6}. Conversely, suppose that $J$ does not satisfy the (PS)$_{\alpha _{s}(\Omega )}-$condition. By Proposition \ref{c2}, there exists a (PS)$_{\alpha _{s}(\Omega) }$-sequence $\{u_n\}$ in $H_{s}(\Omega)$ for $J$ such that $\left\{ \xi_{n}u_{n}\right\}$ is also a (PS)$_{\alpha _{s}(\Omega) }$-sequence in $H_{s}(\Omega)$ for $J$, where $\xi_{n}$ is as in (\ref{11}). Let $v_{n}=\xi _{n}u_{n}$, we obtain $$\begin{gathered} J(v_{n})=\alpha _{s}(\Omega )+\mathrm{o}(1)\text{,} \\ J'(v_{n})=\mathrm{o}(1)\quad\text{in }H^{-1}(\Omega) . \end{gathered} \label{7}$$ Since $\Omega$ is a $y$-symmetric large domain in $\mathbf{A}^{r}$, there exists a $n_{0}\in \mathbb{N}$ such that $v_{n}=0$ in $\overline{\Omega _{n_{0}}% }$ for $n>2n_{0}$, and two disjoint subdomains $\Omega _{1}$ and $\Omega _{2}$ such that \begin{gather*} (x,y)\in \Omega _{2}\quad\text{if and only if}\quad (x,-y)\in \Omega _{1}, \\ \Omega \ \backslash \ \overline{\Omega _{n_{0}}}=\Omega _{1}\cup \Omega _{2}, \end{gather*} where $\Omega _{n}=\left\{ z\in \Omega :-n0$ such that if $u\in \mathbf{M}% _{0}(\mathbf{S})$ and $J(u) \leq \alpha (\mathbf{A}^{r}) +\delta (\varepsilon ,l)$, then either $\int_{\mathbf{S}_{-l}^{+}}| u| ^{p}<\varepsilon$ or $\int_{\mathbf{S}_{l}^{-}}|u| ^{p}<\varepsilon$. \end{lemma} \begin{proof} We divide the proof into the following steps: \noindent Step 1: Suppose that there exist $c>0,l_{0}\geq 0$ and $\{u_n\} \subset \mathbf{M}_{0}(\mathbf{S})$ such that \begin{gather} J(u_{n}) =\alpha (\mathbf{A}^{r}) +o(1) \,, \\ \int_{\mathbf{S}_{-l_{0}}^{+}}| u_{n}| ^{p}\geq c\,, \label{27}\\ \int_{\mathbf{S}_{l_{0}}^{-}}| u_{n}| ^{p}\geq c\,. \label{28} \end{gather} From Lemma \ref{p7}, $\{u_n\}$ is a (PS)$_{\alpha (\mathbf{A}^{r})}$-sequence in $H_{0}^{1}(\mathbf{S})$ for $J$. Since $\mathbf{S}$ is a proper large domain in $\mathbf{A}^{r}$, by Proposition \ref {c1} and Lemma \ref{e4}, there exists a subsequence $\{u_n\}$ such that $\{ \xi_{n}u_{n}\}$ is also a (PS)$_{\alpha (\mathbf{S}) }$-sequence in $H_{0}^{1}(\mathbf{S})$ for $J$, where $\xi _{n}$ is as in (\ref{11}). Let $v_{n}=\xi _{n}u_{n}$, we obtain $$\begin{gathered} J(v_{n}) = \alpha (\mathbf{A}^{r}) +o(1), \\ J'(v_{n}) = o(1)\quad\text{in }H^{-1}(\mathbf{S}), \end{gathered} \label{29}$$ and there exists a $n_{0}>l_{0}$ such that $v_{n}=0$ in $\overline{\mathbf{A}% (n_{0}) }$ for $n>2n_{0}$, where $\mathbf{A}(n) =% \mathbf{S}_{-n,n}$. Moreover, $v_{n}=v_{n}^{+}+v_{n}^{-}$ and $v_{n}^{\pm }(z)=\begin{cases} v_{n}(z) & \text{for }z\in \mathbf{S}_{\mp l_{0}}^{\pm }, \\ 0 & \text{for }z\notin \mathbf{S}_{\mp l_{0}}^{\pm }. \end{cases}$ Then $v_{n}^{\pm }\in H_{0}^{1}(\mathbf{S}_{\mp l_{0}}^{\pm })$ and $a(v_{n}^{\pm }) =b(v_{n}^{\pm }) +o(1)$. By (\ref{29}), we obtain $J'(v_{n}^{\pm })=o(1)\quad \text{strongly in }H^{-1}(\mathbf{S}_{\mp l_{0}}^{\pm }).$ Thus, $\alpha (\mathbf{A}^{r}) +o(1) =J(v_{n}) =J(v_{n}^{+}) +J(v_{n}^{-}).$ Assume that $J(v_{n}^{\pm })=c^{\pm }+o(1)$. Then $$c^{+}+c^{-}=\alpha (\mathbf{A}^{r}) . \label{30}$$ Since $c^{\pm }$ are (PS)-values in $H_{0}^{1}(\mathbf{S}_{\mp l_{0}}^{\pm })$ for $J$, they are nonnegative. Moreover, the half strips $\mathbf{S}_{-l_{0}}^{+}$ and $\mathbf{S}_{l_{0}}^{-}$ are proper large domains in $\mathbf{A}^{r}$, From Lemma \ref{e4}, we have $$\alpha (\mathbf{A}^{r})=\alpha (\mathbf{S}_{-l_{0}}^{+})=\alpha (\mathbf{S}% _{l_{0}}^{-}). \label{31}$$ Thus, by $(\ref{30})$, $(\ref{31})$ and the definition of Nehari minimization problem, we may assume that $c^{+}=\alpha (% \mathbf{S}_{-l_{0}}^{+})=\alpha (\mathbf{A}^{r})$ and $c^{-}=0$. Next, for $n>2n_{0}$, \begin{align*} \int_{\mathbf{S}}|u_n| ^{p} &= \int_{\mathbf{S}}|v_n| ^{p}+o(1) \\ &= \int_{\mathbf{S}_{-l_{0}}^{+}}| v_{n}^{+}| ^{p} +\int_{\mathbf{S}_{l_{0}}^{-}}|u_n| ^{p}+o(1) . \end{align*} Thus, \begin{align*} \int_{\mathbf{S}_{l_{0}}^{-}}|u_n| ^{p} &=\int_{\mathbf{S}}|u_n| ^{p}-\int_{\mathbf{S}_{-l_{0}}^{+}}|v_{n}^{+}| ^{p}+o(1) \\ &=(\frac{2p}{p-2}) \alpha (\mathbf{A}^{r})-(\frac{2p}{p-2}% ) \alpha (\mathbf{A}^{r}) +o(1) \\ &=o(1) , \end{align*} which contradicts to $(\ref{28})$. \noindent Step 2: Suppose that there exists a $u_{0}\in \mathbf{M}_{0}(\mathbf{S})$ with $J(u_{0}) <\alpha (\mathbf{A}^{r}) +\delta (\varepsilon )$ such that $\int_{\mathbf{S}_{-l_{0}}^{+}}| u_{0}| ^{p}<\varepsilon \quad\text{and}\quad \int_{\mathbf{S}_{l_{0}}^{-}}| u_{0}| ^{p}<\varepsilon .$ Then \begin{align*} \frac{2p}{(p-2) }\alpha (\mathbf{S}) &\leq \int_{\mathbf{S}}| u_{0}| ^{p}=\int_{\mathbf{S}_{l_{0}}^{+}}| u_{0}| ^{p}+\int_{\mathbf{S}_{l_{0}}^{-}}| u_{0}| ^{p} \\ &< \frac{p}{(p-2) }\alpha (\mathbf{A}^{r}) +\frac{p}{% (p-2) }\alpha (\mathbf{A}^{r}) \\ &= \frac{2p}{(p-2) }\alpha (\mathbf{A}^{r}) , \end{align*} which is also a contradiction. \end{proof} \begin{lemma} \label{m2} If $\alpha _{s}(\mathbf{S}) <2\alpha (\mathbf{A}^{r})$. Then for each $0<\varepsilon \leq (\frac{p}{p-2}) \alpha (\mathbf{A}^{r})$, there exist positive numbers $l(\varepsilon )$ and $\delta (\varepsilon)$ such that if $u\in \mathbf{M}_{s}(\mathbf{S})$ and $J(u) <\alpha _{s}(\mathbf{S}) +\delta (\varepsilon )$, then $\int_{(\mathbf{S}_{-l(\varepsilon ) ,l(\varepsilon) }) ^{c}}|u| ^{p}<\varepsilon$. \end{lemma} \begin{proof} If not, there exist a positive number $c\leq (\frac{p}{p-2}) \alpha (\mathbf{A}^{r})$ and $\{u_n\} \subset \mathbf{M}_{s}(\mathbf{S})$ such that $$\begin{gathered} J(u_{n}) =\alpha _{s}(\mathbf{S}) +\frac{1}{n}\,,\\ \int_{(\mathbf{S}_{-n,n}) ^{c}}|u_n| ^{p}\geq c% \quad\text{for all }n=1,2,\ldots . \end{gathered} \label{16}$$ By Lemma \ref{p7}, $\{u_n\}$ is a (PS)$_{\alpha _{s}(\mathbf{S}) }$-sequence in $H_{s}(\mathbf{S})$ for $J$. Since $\alpha _{s}(\mathbf{S}) <2\alpha (\mathbf{A% }^{r})$. By Proposition \ref{e7}, $J$ is satisfying (PS)$_{\alpha _{s}(\mathbf{S}) }-$condition in $H_{s}(\mathbf{S})$. Thus, there exist a subsequence $\{u_n\}$ and $u_{0}\in H_{s}(\mathbf{S})$ such that $u_{n}\to u_{0}\quad\text{strongly in }H_{s}(\mathbf{S}) .$ By the Sobolev imbedding theorem and the Vitali convergence theorem, there exists a $l_{0}>0$ such that $\int_{(\mathbf{S}_{-l_{0},l_{0}}) ^{c}}|u_n| ^{p}<% \frac{c}{2}\quad\text{for all }n,$ which contradicts to $(\ref{16})$. \end{proof} \begin{lemma}\label{m3} Suppose that the equation $(\ref{E1})$ in $\mathbf{S}$ does not admit any solution $u_{0}$ such that $J(u_{0}) =\alpha _{s}(\mathbf{S})$. Then for each positive number $\varepsilon \leq (\frac{2p}{p-2}) \alpha _{s}(\mathbf{S})$ and $% l$, there exists a $\delta (\varepsilon ,l) >0$ such that if $u\in \mathbf{M}_{s}(\mathbf{S})$ and $J(u) <\alpha _{s}(\mathbf{S}) +\delta (\varepsilon ,l)$, then $% \int_{\mathbf{S}_{-l,l}}|u| ^{p}<\varepsilon$. \end{lemma} The proof of this lemma is similar to the proof of Lemma \ref{m1}, and is omitted here. For $\Theta _{t}=\mathbf{A}_{-t,t}^{r}\backslash \overline{\omega }$, consider the filtration of $J$ in $\mathbf{M}(\Theta_{t})$, $F(\Theta _{t}) =\left\{ u\in \mathbf{M}_{0}(\Theta _{t}) :J(u) \leq \alpha _{s}(\mathbf{S}) \right\} .$ Note that if $F(\Theta _{t})$ is a nonempty set, then $\alpha (\Theta _{t}) =\inf_{v\in F(\Theta _{t}) }J(v) .$ Note that $\Theta _{t_{1}}\subset \Theta _{t_{2}}$ for $t_{1}\alpha _{X}(\Theta _{t_{2}})$ for $t_{1}0$ such that $$\alpha (\mathbf{S}) <\alpha (\Theta _{t}) \leq \alpha _{s}(\mathbf{S}) \quad\text{for all }t\geq t_{0}. \label{20}$$ Since $\Theta_{t}$ is a $y$-symmetric bounded domain, by Lemma \ref{c8}, $F(\Theta _{t})$ is nonempty for all $t\geq t_{0}$. Moreover, $\alpha _{s}(\Theta _{t}) =\inf_{v\in \mathbf{M}_{s}(\Theta _{t}) }J(v)$ and $$\alpha _{s}(\mathbf{S}) <\alpha _{s}(\Theta _{t}) \quad\text{for all }t>0. \label{21}$$ We can conclude that $F(\Theta _{t}) \cap \mathbf{M}_{s}(\Theta _{t}) =\phi$ for all $t\geq t_{0}$. By $(\ref{20})$, $(\ref{21})$ and Lemma \ref{c8}, we have $$\alpha (\Theta _{t}) \leq \alpha _{s}(\mathbf{S}) <\alpha _{s}(\Theta _{t}) \quad\text{for all } t\geq t_{0} \label{22}$$ and the equation $(\ref{E1})$ in $\Theta _{t}$ admit two disjoint positive solutions $u_{1},u_{2}$ such that $J(u_{1}) =\alpha _{s}(\Theta _{t})$ and $J(u_{2}) =\alpha (\Theta _{t})$. Take $u_{3}(x,y) =u_{2}(x,-y)$, then $J(u_{3}) =\alpha (\Theta _{t})$% , $u_{3}\in \mathbf{M}_{0}(\Theta _{t})$ and $u_{3}$ is third positive solution. \end{proof} \begin{remark} \rm By Theorem \ref{t1}, there exists a $t_{0}>0$ such that for $t\geq t_{0}$, the equation $(\ref{E1})$ in $\Theta _{t}$ has one $y-$% symmetric positive solution $u_{1}$ and two non-axially symmetric positive solutions $u_{2}$ and $u_{3}$. Moreover, $\int_{\Theta _{t}}|u_1| ^{p}=\frac{2p}{p-2}\alpha _{s}(\Theta _{t}) >\frac{2p}{p-2}\alpha _{s}(\mathbf{S})$ and $\int_{\Theta _{t}}| u_{i}| ^{p}=\frac{2p}{p-2}\alpha (\Theta _{t}) \leq \frac{2p}{p-2}\alpha _{s}(\mathbf{S}) \quad\text{for }i=2,3.$ Thus, we can conclude that \begin{gather*} \int_{\Theta _{t}^{+}}|u_1| ^{p}=\int_{\Theta _{t}^{-}}|u_1| ^{p}>\frac{p}{p-2}\alpha _{s}(\mathbf{S}) ,\\ \int_{\Theta _{t}^{+}}|u_2| ^{p}\leq \frac{p}{p-2}\alpha _{s}(\mathbf{S})\\ \int_{\Theta _{t}^{-}}|u_3| ^{p}\leq \frac{p}{p-2}\alpha _{s}(\mathbf{S}) , \end{gather*} where $\Theta _{t}^{+}=\left\{ (x,y) \in \Theta _{t}:y\geq 0\right\}$ and $\Theta _{t}^{-}=\left\{ (x,y) \in \Theta _{t}:y\leq 0\right\}$. \end{remark} Next, we describe the concentration of solutions of equation $% (\ref{E1})$ in $\Theta _{t}$. We need the following notation: \begin{gather*} \Theta _{t}(-l,l) =\left\{ (x,y) \in \Theta_{t}:-l\leq y\leq l\right\} ; \\ \Theta _{t}^{+}(l) =\left\{ (x,y) \in \Theta_{t}:y\geq l\right\} ; \\ \Theta _{t}^{-}(l) =\left\{ (x,y) \in \Theta_{t}:y\leq l\right\} . \end{gather*} Then we have the following results. \begin{theorem} \label{t2} Suppose that $\alpha _{s}(\mathbf{S})<2\alpha (\mathbf{A}^{r})$. Then for each positive number $\varepsilon \leq (\frac{p}{p-2}) \alpha (\mathbf{A}^{r})$, there exist positive numbers $t_{0}>l_{0}$ such that for $t>t_{0}$ the equation $% (\ref{E1})$ in $\Theta _{t}$ has three positive solutions $u_{1}$, $u_{2}$ and $u_{3}$. Moreover,\begin{itemize} \item[(i)] $\int_{(\Theta _{t}(-l_{0},l_{0}) )^{c}}|u_1| ^{p}<\varepsilon$ \item[(ii)] $\int_{\Theta _{t}^{+}(-l_{0}) }|u_2| ^{p}<\varepsilon$ and $\int_{\Theta _{t}^{-}(l_{0}) }|u_3| ^{p}<\varepsilon$. \end{itemize} \end{theorem} \begin{proof} Since $\alpha _{s}(\mathbf{S}) <2\alpha (\mathbf{A}^{r})$. By Lemma \ref{m2}, for each positive number $\varepsilon \leq (\frac{p}{p-2}) \alpha (\mathbf{A}^{r})$, there exist positive numbers $l_{0}$ and $\delta (\varepsilon )$ such that if $u\in \mathbf{M}_{s}(\mathbf{S})$ and $J(u) <\alpha _{s}(\mathbf{S}) +\delta (\varepsilon )$, then $\int_{(\mathbf{S}_{-l_{0},l_{0}}) ^{c}}| u| ^{p}<\varepsilon$. Moreover, by Lemma \ref{m4}, there exists a $t_{1}>0$ such that $\alpha _{s}(\Theta _{t}) <\alpha _{s}(\mathbf{S}% ) +\delta (\varepsilon )$ for all $t>t_{1}$. Since $% \Theta _{t}$ is a bounded domain, by Lemma \ref{c8}, the equation $(\ref{E1})$ in $\Theta _{t}$ admits a positive solution $u_{1}\in H_{0}^{1}(\Theta _{t})$ such that $J(u_{1}) =\alpha _{s}(\Theta _{t})$. Thus, $u_{1}\in \mathbf{M}_{s}(\mathbf{S})$, \begin{gather*} J(u_{1}) <\alpha (\mathbf{A}^{r}) +\delta (\varepsilon )\,,\\ \int_{(\mathbf{S}_{-l_{0},l_{0}}) ^{c}}|u_1| ^{p} =\int_{(\Theta _{t}(-l_{0},l_{0}) ) ^{c}}|u_1|^{p}<\varepsilon . \end{gather*} Fixed the positive numbers $\varepsilon ,l_{0}$. By Lemma \ref{m1}, there exists a $\delta (\varepsilon ,l_{0}) >0$ such that if $u\in \mathbf{M} _{0}(\mathbf{S})$ and $J(u) <\alpha (\mathbf{A }^{r}) +\delta (\varepsilon ,l_{0})$, then $\int_{\mathbf{S }_{-l_{0}}^{+}}|u| ^{p}<\varepsilon$ or $\int_{\mathbf{S} _{l_{0}}^{-}}|u| ^{p}<\varepsilon$. Moreover, by Lemma \ref{m4}, there exists a $t_{2}>0$ such that $\alpha (\Theta _{t}) <\alpha (\mathbf{A}^{r}) +\delta (\varepsilon )$ for all $t>t_{2}$. Since $\Theta _{t}$ is a bounded domain, by Lemma \ref{c8}, the equation $(\ref{E1})$ in $\Theta _{t}$ admits a positive solution $u_{2}$ such that $J(u_{2}) =\alpha (\Theta _{t})$. Then $u_{2}\in \mathbf{M}_{0}(\Theta _{t}) \subset \mathbf{M}_{0}(\mathbf{S}) ,\;J(u_{2}) <\alpha (\mathbf{A}^{r}) +\delta (\varepsilon )$ and either $$\int_{\Theta _{t}^{+}(-l_{0}) }|u_2| ^{p}<\varepsilon \;\text{or\ }\int_{\Theta _{t}^{-}(l_{0}) }|u_2| ^{p}<\varepsilon . \label{23}$$ Without loss of generality, we may assume that $\int_{\Theta _{t}^{+}(-l_{0}) }|u_2|^{p}<\varepsilon .$ Take $u_{3}(x,y) =u_{2}(x,-y)$, then $u_{3}$ is third positive solution and $\int_{\Theta _{t}^{-}(l_{0}) }|u_3| ^{p}<\varepsilon .$ Now, let $t_{0}=\max \{ t_{1},t_{2}\}$. Since $\varepsilon \leq (\frac{p}{p-2}) \alpha (\mathbf{A}^{r})$, $u_{i}$ is disjoint for $i=1,2,3$. \end{proof} \begin{theorem}\label{t3} Suppose that the equation $(\ref{E1})$ in $\mathbf{S}$ does not admit any solution $u_{0}$ such that $J(u_{0}) =\alpha _{s}(\mathbf{S})$. Then for positive numbers $\varepsilon \leq (\frac{p}{p-2}) \alpha (\mathbf{A}^{r})$ and $l$, there exists a positive number $t_{0}$ such that for $t>t_{0}$, the equation $(\ref{E1})$ in $\Theta _{t}$ has three positive solutions $u_{1}, u_{2}$ and $u_{3}$. Moreover, \begin{itemize} \item[(i)] $\int_{(\Theta_{t}(-l,l) ) ^{c}}|u_1|^{p}<\varepsilon$ \item[(ii)] $\int_{\Theta_{t}^{+}(-l) }|u_2| ^{p}<\varepsilon$ and $\int_{\Theta _{t}^{-}(l) }|u_3|^{p}<\varepsilon$. \end{itemize} \end{theorem} The proof of this theorem is similar to the proof of Theorem \ref{t2} and therefore omitted here. Note that if $u_{1},u_{2}$ and $u_{3}$ are positive solutions as in Theorem \ref{t2} or Theorem \ref{t3}, then $u_{1}$ is $y$-symmetric and $u_{2},u_{3}$ are non-axially symmetric. \section{Dynamic System of Solutions} For $m=1,2,\cdots$, define $\Theta _{m}=\mathbf{A}_{-m,m}^{r}\backslash \overline{\omega }$, then $\{\Theta _{m}\}$ is an increasing sequence and $\mathbf{S}=\mathbf{A}^{r}\backslash \overline{\omega } =\cup_{m=1}^\infty \Theta _{m}.$ By Theorem \ref{t1}, there exists a $t_{0}>0$ such that for $m\geq t_{0}$, the equation $(\ref{E1})$ in $\Theta _{m}$ admit one $y-$% symmetric positive solution $u_{m}^{1}$ and two non-axially symmetric positive solutions $u_{m}^{2}$ and $u_{m}^{3}$. Note that $J(u_{m}^{2}) =J(u_{m}^{3}) =\alpha (\Theta _{m}) <\alpha _{s}(\Theta _{m}) =J(u_{m}^{1}) \quad\text{for all }m\geq t_{0}.$ Then we have the following results. \begin{theorem} \label{t4} \begin{itemize} \item[(i)] The sequence $\{ u_{m}^{1}\}$ is a (PS)$_{\alpha _{s}(\mathbf{S}) }$-sequence in $H_{s}(\mathbf{S})$ for $J$ \item[(ii)] If $\alpha _{s}(\Theta _{m_{0}}) <2\alpha(\mathbf{A}^{r})$ for some $m_{0}>0$, then there exist a subsequence $u_{m}^{1}$ and $u^{1}\in H_{s}(\Omega)$ such that $u_{m}^{1}\to u^{1}$ strongly in $L^{p}(\mathbf{S} )$ in $H_{s}(\mathbf{S} )$ as $m\to \infty$ and $J(u^{1}) =\alpha _{s}(\mathbf{S} )$ \item[(iii)] If the equation $(\ref{E1})$ in $\mathbf{S% }$ does not admit any solution $u_{0}$ such that $J(u_{0}) =\alpha _{s}(\mathbf{S})$, then $u_{m}^{1}\rightharpoonup 0$ weakly in $L^{p}(\mathbf{S})$ and in $H_{0}^{1}(\mathbf{S})$ as $m\to \infty$. \end{itemize} \end{theorem} \begin{proof} (i) By Lemma \ref{m4}, we have $J(u_{m}^{1}) =\alpha _{s}(\Theta _{m}) =\alpha _{s}(\mathbf{S}) +o(1)$. Since $u_{m}^{1}\in \mathbf{M}_{s}(\Theta _{m}) \subset \mathbf{M}_{s}(\mathbf{S})$, from Lemma \ref {p7} we can conclude that $\{ u_{m}^{1}\}$ is a (PS)$_{\alpha _{s}(\mathbf{S}) }$-sequence in $H_{s}(\mathbf{S})$ for $J$. \noindent (ii) Since $\alpha _{s}(\Theta _{m_{0}}) <2\alpha (\mathbf{A}^{r})$ for some $m_{0}>0$ and $\Theta _{m}\subset \Theta _{m+1}\subset \mathbf{S}$ for each $m$, we have $\alpha _{s}(\mathbf{S}) <2\alpha (\mathbf{A}^{r})$. By Proposition \ref{e7}, $J$ satisfies the (PS)$_{\alpha _{s}(\mathbf{S})}-$condition in $H_{s}(\mathbf{S})$. Then there exist a subsequence $\left\{ u_{m}^{1}\right\}$ and a $y$-symmetric positive solution $u^{1}$ of equation $(\ref{E1})$ in $\mathbf{S}$ such that $u_{m}^{1}\to u^{1}$ strongly in $L^{p}(\mathbf{S})$ and in $H_{s}(\mathbf{S})$ and $J(u^{1}) =\alpha _{s}(\mathbf{S})$. \noindent (iii) Let $v\in L^{q}(\mathbf{S})$, where $\frac{1}{p}+\frac{1}{q}=1$. Then for each $\varepsilon >0$, there exists a $l>0$ such that $\int_{(\mathbf{S}_{-l,l}) ^{c}}| v| ^{q}<\varepsilon ^{q}.$ Moreover, by Theorem \ref{t3}, there exists a $m_{0}$ such that $\int_{\mathbf{S}_{-l,l}}| u_{m}^{1}| ^{q}<\varepsilon ^{p} \quad\text{for all }m>m_{0}.$ Thus, for each $\varepsilon >0$, there exists a $m_{0}$ such that \begin{align*} \int_{\mathbf{S}}u_{m}^{1}v &= \int_{(\mathbf{S}_{-l,l})^{c}}u_{m}^{1}v+\int_{\mathbf{S}_{-l,l}}u_{m}^{1}v \\ &\leq \Big(\int_{(\mathbf{S}_{-l,l}) ^{c}}|u_{m}^{1}| ^{p}\Big) ^{1/p} \Big(\int_{(\mathbf{S}_{-l,l}) ^{c}}| v| ^{q}\Big) ^{\frac{1}{q}} +\Big(\int_{\mathbf{S}_{-l,l}}| u_{m}^{1}| ^{p}\Big) ^{1/p} \Big(\int_{\mathbf{S}_{-l,l}}| v| ^{q}\Big) ^{\frac{1}{q}} \\ &\leq (c_{1}+c_{2}) \varepsilon \quad\text{for all }m>m_{0}, \end{align*} where $c_{1}=(\frac{2p}{p-2}\alpha _{s}(\Theta _{1}) )$ and $c_{2}=\left\| v\right\| _{L^{q}}$. This implies $u_{m}^{1}\rightharpoonup 0$ weakly in $L^{p}(\mathbf{S})$ as $m\to \infty$. Since $u_{m}^{1}$ is a solution of equation $(\ref{E1})$ in $\Theta _{m}$, we have $\int_{\Theta _{m}}\nabla u_{m}^{1}\nabla \varphi +u_{m}^{1}\varphi =\int_{\Theta _{m}}| u_{m}^{1}| ^{p-2}u_{m}^{1}\varphi \quad\text{for all }\varphi \in H_{0}^{1}(\Theta _{m}) .$ First, we need to show for each $\varepsilon >0$ and $\varphi \in C_{c}^{1}(\mathbf{S})$, there exists $m_{0}$ such that $\int_{\Theta _{m}}\nabla u_{m}^{1}\nabla \varphi +u_{m}^{1}\varphi <\varepsilon \quad\text{for all }m>m_{0}.$ For $\varphi \in C_{c}^{1}(\mathbf{S})$. Let $K=\mathop{\rm supp}\varphi$, then $K\subset \mathbf{S}$ is compact and there exists a $m_{1}$ such that $K\subset \Theta _{m}$ for all $m\geq m_{1}$. Thus, by Theorem \ref {t3} for each $\varepsilon >0$, there exist $l_{0}>0$ and $m_{0}$ such that $% \varphi \in H_{0}^{1}(\Theta _{m})$, \begin{gather*} \int_{(\mathbf{S}_{-l_{0},l_{0}}) ^{c}}|\varphi|^{p}=0\,,\\ \int_{\mathbf{S}_{-l_{0},l_{0}}}| u_{m}^{1}| ^{p} <\varepsilon ^{\frac{p-1}{p}}\quad\text{for all }m>m_{0}. \end{gather*} We obtain \begin{align*} \int_{\Theta _{m}}| u_{m}^{1}| ^{p-2}u_{m}^{1}\varphi &=\int_{(\mathbf{S}_{-l_{0},l_{0}}) ^{c}}| u_{m}^{1}| ^{p-2}u_{m}^{1}\varphi +\int_{\mathbf{S}_{-l,l_{0}}}| u_{m}^{1}| ^{p-2}u_{m}^{1}\varphi \\ &\leq \Big(\int_{(\mathbf{S}_{-l_{0},l_{0}}) ^{c}}| u_{m}^{1}| ^{p}\Big) ^{\frac{p-1}{p}}\Big(\int_{(\mathbf{S} _{-l_{0},l_{0}}) ^{c}}| \varphi | ^{p}\Big) ^{1/p} \\ &\quad+\Big(\int_{\mathbf{S}_{-l_{0},l_{0}}}| u_{m}^{1}| ^{p}\Big) ^{\frac{p-1}{p}}\Big(\int_{\mathbf{S}_{-l_{0},l_{0}}}| \varphi | ^{p}\Big) ^{1/p} \leq c\varepsilon \end{align*} and \begin{align*} \int_{\mathbf{S}}\nabla u_{m}^{1}\nabla \varphi +\int_{\mathbf{S}% }u_{m}^{1}\varphi &=\int_{\Theta _{m}}\nabla u_{m}^{1}\nabla \varphi +\int_{\Theta _{m}}u_{m}^{1}\varphi \\ &=\int_{\Theta _{m}}| u_{m}^{1}| ^{p-2}u_{m}^{1}\varphi \quad\text{for all }m>m_{0}. \end{align*} This follows that $$\int_{\mathbf{S}}\nabla u_{m}^{1}\nabla \varphi +\int_{\mathbf{S}% }u_{m}^{1}\varphi \leq c\varepsilon \quad\text{for all }m>m_{0}. \label{36}$$ Since $\alpha _{s}(\Theta _{m+1}) <\alpha _{s}(\Theta )$, there exists a $C>0$ such that $\Vert u_{m}^{1}\Vert _{H^{1}}\leq C$. Thus, for each $\varepsilon >0$ and $\psi \in H_{0}^{1}(\mathbf{S})$, there exists a $\varphi \in C_{c}^{1}(\mathbf{S})$ such that $$\Vert \psi -\varphi \Vert _{H^{1}}<\frac{\varepsilon }{C}. \label{37}$$ From $(\ref{36})$ and $(\ref{37})$, we can conclude that for each $\varepsilon >0$ and $\psi \in H_{0}^{1}(\mathbf{S})$, there exists a $m_{0}>0$ such that \begin{align*} \left\langle u_{m}^{1},\psi \right\rangle _{H^{1}} &=\left\langle u_{m}^{1},\psi -\varphi \right\rangle _{H^{1}} +\left\langle u_{m}^{1},\varphi \right\rangle _{H^{1}} \\ &\leq C\Vert \psi -\varphi \Vert _{H^{1}}+\left\langle u_{m}^{1},\varphi \right\rangle _{H^{1}} \\ &< \varepsilon +c\varepsilon \quad\text{for }m>m_{0}. \end{align*} This implies $u_{m}^{1}\rightharpoonup 0$ weakly in $H_{0}^{1}(\mathbf{S})$. \end{proof} \begin{theorem} \label{t5} \begin{itemize} \item[(i)] The sequence $\{ u_{n}^{i}\}$ is a (PS)$_{\alpha (\mathbf{S}) }$-sequence in $H_{0}^{1}(\mathbf{S})$ for $J$, for $i=2,3$ \item[(ii)] $u_{n}^{i}\rightharpoonup 0$ weakly in $L^{p}(\mathbf{S})$ and in $H_{0}^{1}(\mathbf{S})$ as $n\to \infty$, for $i=2,3$. \end{itemize} \end{theorem} The proof of this theorem is similar to the proof of Theorem \ref{t4} (i) and (iii). \begin{thebibliography}{99} \bibitem{BL} A. Bahri and L. P. Lions, \textit{On the existence of positive solutions of semilinear elliptic equations in unbounded domains}, Ann. I. H. P. Analyse non lineaire \textbf{14} (1997), 365-413. \bibitem{By1} J. Byeon, \textit{Existence of large positive solutions of some nonlinear elliptic equations on singularly perturbed domains,} Commun. in Partial Differential Equations \textbf{22}, (1997) 1731-1769. \bibitem{By2} J. Byeon, \textit{Nonlinear elliptic problems on singularly perturbed domains,} Proc. Royal Society Edinburgh sect. A \textbf{131}, (2001) 1023-1037. \bibitem{CNZ} G. Chen, W. M. Ni, and J. Zhou, \textit{Algorithms and visualization for solution of nonlinear elliptic problems,} International Journal of Bifurcation and Chaos \textbf{10}, (2000) 1565-1612. \bibitem{Da} E. N. Dancer, \textit{The effect of domain shape on the number of positive solution of certain nonlinear equations}, J. of Diff. Equation \textbf{74}, (1988) 120-156. \bibitem{GNN} B. Gidas, W. M. Ni, and L. Nirenberg, \textit{Symmetry and related properties via the maximum principle}, Comm. Math. Phys. \textbf{68}, (1978) 209-243. \bibitem{LTW} W. C. Lien, S. Y. Tzeng, and H. C. Wang, \textit{Existence of solutions of semilinear elliptic problems in unbounded domains}, Differential Integral Equations \textbf{6}, (1993) 1281-1298. \bibitem{P} R. Palais, \textit{The Principle of symmetric criticality,} Comm. Math. Phys. 69, (1979) 19-30. \bibitem{R} P. H. Rabinowitz, \textit{Minimax Methods in Critical Point Theory with Applications to Differential Equations,} Regional Conference Series in Mathematics, American Mathematical Society, 1986. \bibitem{WW} H. C. Wang and T. F. Wu, \textit{Symmetric Breaking in a Bounded Symmetric Domain}, NoDEA-Nonlinear Differential Equations Appl., to appear. \bibitem{Wi} M. Willem, \textit{Minimax Theorems}, Birkh\"{a}user, Boston, 1996. \end{thebibliography} \end{document}