\documentclass[reqno]{amsart} \usepackage{amssymb} \usepackage{graphicx} \pdfoutput=1\relax\pdfpagewidth=8.26in\pdfpageheight=11.69in\pdfcompresslevel=9 \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Monograph 06, 2004, (142 pages).\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2004 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2004/Mon. 06\hfil Palais-Smale approaches] {Palais-Smale approaches to semilinear elliptic equations in unbounded domains} \author[Hwai-chiuan Wang\hfil EJDE-2004/Mon. 06\hfilneg] {Hwai-chiuan Wang} \address{Department of Mathematics\\ National Tsing Hua University\\ Hsinchu, Taiwan} \email{hwang@mail.math.nthu.edu.tw} \date{} \thanks{Submitted September 17, 2004. Published September 30, 2004.} \subjclass[2000]{35J20, 35J25} \keywords{Palais-Smale condition; index; decomposition theorem;\hfill\break\indent achieved domain; Esteban-Lions domain; symmtric Palais-Smale condition} \begin{abstract} Let $\Omega$ be a domain in $\mathbb{R}^{N}$, $N\geq1$, and $2^{\ast}=\infty$ if $N=1,2$, $2^{\ast}=\frac{2N}{N-2}$ if $N>2$, $22$, $2|x|^{2}\};\\ \mathbf{P}^{-}=\{(x,-y): (x,y)\in\mathbf{P}^{+}\};\\ \mathbf{C}=\{(x,y)\in\mathbb{R}^{N}: |x|0$, $z\in\Omega$ exists such that $B(z;r)\subset\Omega$; \newline $(i')$ We say that $\Omega$ is a strictly large domain in $\mathbb{R}^{N}$ if $\Omega$ contains an infinite cone of $\mathbb{R}^{N}$; \newline $(ii)$ We call $\Omega$ a large domain in $\mathbf{A}^{r}$ if for any positive number $m$, $a$, $b$ exist such that $b-a=m$ and $\mathbf{A} _{a,b}^{r}\subset\Omega$; \newline $(ii')$ We call $\Omega$ a strictly large domain in $\mathbf{A}^{r} $ if $\Omega$ contains a semi-strip of $\mathbf{A}^{r}$; \newline $(iii)$ We call $\Omega$ a large domain in $\mathbf{A}^{r_{1},r_{2}}$ if for any positive number $m$, $a$, $b$ exist with $a0$, $\varphi\in H^{1}(\mathbb{R}^{N}) $, and $\phi\in C_{c}^{1}(\mathbb{R}^{N}) $ exist such that \[ \Vert\varphi-\phi\Vert_{H^{1}}<\varepsilon/2(\Vert u\Vert_{H^{1}}+1). \] Let $K=\mathop{\rm supp}\phi$, then $K$ is compact. We have \begin{align*} \langle u(z+z_{n}),\phi(z) \rangle _{H^{1}} &={\int_{\mathbb{R}^{N}}} \nabla u(z+z_{n})\nabla\phi(z)dz+ {\int_{\mathbb{R}^{N}}}u(z+z_{n})\phi(z)dz\\ &={\int_{K}}\nabla u(z+z_{n})\nabla\phi(z)dz+ {\int_{K}} u(z+z_{n})\phi(z)dz\\ &\leq\| \nabla u(z+z_{n})\| _{L^{2}(K)}\| \nabla\phi\| _{L^{2}(K)}+\| u(z+z_{n})\| _{L^{2} (K)}\| \phi\| _{L^{2}(K)}\\ &= o(1)\quad \text{as } n\to\infty. \end{align*} Thus, for some $N>0$ such that $| \langle u(z+z_{n}),\phi (z)\rangle _{H^{1}}| <\frac{\varepsilon}{2}$ for $n\geq N$. In addition, \begin{align*} \langle u(z+z_{n}),\varphi(z)\rangle _{H^{1}} & =\langle u(z+z_{n}),\varphi(z)-\phi(z)\rangle _{H^{1}}+\langle u(z+z_{n}),\phi(z)\rangle _{H^{1}}\\ & \leq\Vert u(z+z_{n})\Vert_{H^{1}(\mathbb{R}^{N}) }\Vert \varphi(z)-\phi(z)\Vert_{H^{1}(\mathbb{R}^{N}) }\\ & \quad +\langle u(z+z_{n}),\phi(z)\rangle _{H^{1}}\\ & \leq\Vert u(z)\Vert_{H^{1}(\mathbb{R}^{N}) }\Vert \varphi(z)-\phi(z)\Vert_{H^{1}(\mathbb{R}^{N}) }+\frac {\varepsilon}{2}\\ & <\varepsilon\text{\ for\ }n\geq N. \end{align*} Therefore, $u(z+z_{n})\rightharpoonup0$ weakly in $H^{1}(\mathbb{R}^{N}) $. \end{proof} \begin{lemma} \label{p6} For $u\in H_{0}^{1}(\mathbf{A}^{r}) $ and $\{z_{n}\}$ in $\mathbf{A}^{r}$ satisfying $| z_{n}| \to\infty$ as $n\to\infty$, then $u(z+z_{n})\rightharpoonup0$ weakly in $H_{0} ^{1}(\mathbf{A}^{r}) $ as $n\to\infty$. \end{lemma} The proof of this lemma is the same as the proof of Lemma \ref{p5}. Therefore, we omit it. Bounded $L^{p}(\Omega)$ sequence admits interesting convergent properties. \begin{lemma}[Br\'{e}zis-Lieb Lemma] \label{p7} Suppose $u_{n}\to u$ a.e. in $\Omega$ and there is a $c>0$ such that $\| u_{n}\| _{L^{p}(\Omega)}\leq c$ for $n=1,2,\dots$. Then \newline $(i)\| u_{n}-u\| _{L^{p}}^{p}=\| u_{n}\| _{L^{p} }^{p}-\| u\| _{L^{p}}^{p}+ o(1)$; \newline$(ii)$ $|u_{n} -u|^{p-2}(u_{n}-u) -|u_{n}|^{p-2}u_{n}+|u|^{p-2}u= o(1)$ in $L^{\frac{p}{p-1}}(\Omega) $. \end{lemma} \begin{proof} $(i)$ Let $\varphi(t) =t^{p}$ for $t>0$, then $\varphi^{\prime }(t) =pt^{p-1}$ and \[ | u_{n}-u| ^{p}-| u_{n}| ^{p}=\varphi(|u_{n} -u|)-\varphi(|u_{n}|)=\varphi'(t) (|u_{n} -u|-|u_{n}|) , \] where $t=(1-\theta)|u_{n}|+\theta|u_{n}-u|\leq|u_{n}|+|u|$ for some $\theta \in[0,1]$. Thus, by the Young inequality, for $\varepsilon>0$ \[ | | u_{n}-u| ^{p}-| u_{n}| ^{p}| \leq p(|u_{n}|+|u|) ^{p-1}|u|\\ \leq d(|u_{n}|^{p-1}|u|) +d|u|^{p}\\ \leq\varepsilon|u_{n}|^{p}+c_{\varepsilon}|u|^{p}. \] Thus, \[ | | u_{n}-u| ^{p}-| u_{n}| ^{p}+|u|^{p} | \leq\varepsilon|u_{n}|^{p}+(c_{\varepsilon}+1)|u|^{p}. \] We have \[ {\int_{\Omega}} | | u_{n}-u| ^{p}-| u_{n}| ^{p}+|u|^{p} | \leq\varepsilon c^{p}+(c_{\varepsilon}+1) {\int_{\Omega}} |u|^{p}. \] Since $\| u\| _{L^{p}}\leq\liminf_{n\to\infty}\| u_{n}\| _{L^{p}}\leq c$. For some $\delta>0$ $|E|<\delta$ implies $ {\int_{E}} |u|^{p}<\varepsilon$. In addition, $K$ in $\mathbb{R}^{N}$ exists such that $|K|<\infty$ and ${\int_{K^{c}}}|u|^{p}<\varepsilon$. Thus, \begin{gather*} {\int_{E}} | | u_{n}-u| ^{p}-| u_{n}| ^{p}+|u|^{p} | \leq(c^{p}+c_{\varepsilon}+1) \varepsilon,\\ {\int_{K^{c}}} | | u_{n}-u| ^{p}-| u_{n}| ^{p}+|u|^{p} | \leq(c^{p}+c_{\varepsilon}+1) \varepsilon. \end{gather*} Clearly, $| | u_{n}-u| ^{p}-| u_{n}| ^{p}+|u|^{p}| = o(1)$ a.e. in $\Omega$. By Theorem \ref{p16} below, ${\int_{\Omega}} | | u_{n}-u| ^{p}-| u_{n}| ^{p}+|u|^{p} | = o(1)$, or \[ \| u_{n}-u\| _{L^{p}}^{p}=\| u_{n}\| _{L^{p}}^{p}-\| u\| _{L^{p}}^{p}+ o(1). \] $(ii)$ Let $\varphi(t)=|t|^{p-2}t$, then $\varphi'(t)=(p-1) |t|^{p-2}$. The proof is similar to part $(i)$ \end{proof} New (PS)-sequences can be produced as follows. \begin{lemma} \label{p8} Let $u_{n}\rightharpoonup u$ weakly in $X(\Omega) $ and \[ J'(u_{n})=-\Delta u_{n}+u_{n}-|u_{n}|^{p-2}u_{n}= o(1)\quad\mbox{in } X^{-1}(\Omega) . \] Then \newline $(i)$ $|u_{n}-u|^{p-2}(u_{n}-u) -|u_{n}|^{p-2} u_{n}+|u|^{p-2}u= o(1)$ in $X^{-1}(\Omega) $; \newline $(ii)$ $J'(\varphi_{n})=-\Delta\varphi_{n}+\varphi_{n}-|\varphi_{n} |^{p-2}\varphi_{n}= o(1)$ in $X^{-1}(\Omega) $ where $\varphi _{n}=u_{n}-u$; \newline $(iii)$ if $\{ u_{n}\} $ is a (PS)$_{\beta}$-sequence, then $\{ \varphi_{n}\} $ is a (PS)$_{(\beta-J(u)) }$-sequence. \end{lemma} \begin{proof} $(i)$ By Lemma \ref{p7}, \[ \int_{\Omega}| |u_{n}-u|^{p-2}(u_{n}-u)-|u_{n}|^{p-2}u_{n} +|u|^{p-2}u| ^{\frac{p}{p-1}}= o(1). \] Now for $\varphi\in H^{1}(\Omega) $, \begin{align*} &| \langle |u_{n}-u|^{p-2}(u_{n}-u)-|u_{n}|^{p-2}u_{n} +|u|^{p-2}u,\varphi\rangle | \\ &=|{\int_{\Omega}} \varepsilon_{n}\varphi|\leq( {\int_{\Omega}}|\varepsilon_{n}|^{\frac{p}{p-1}}) ^{\frac{p-1}{p}}( {\int_{\Omega}} |\varphi|^{p}) ^{1/p}\\ &\leq c\Vert\varepsilon_{n}\Vert_{L^{\frac{p}{p-1}}}\Vert\varphi \Vert_{H^{1}}, \end{align*} where $\varepsilon_{n}=|u_{n}-u|^{p-2}(u_{n}-u)-|u_{n}|^{p-2}u_{n}+|u|^{p-2} u$. Therefore, \[ \| |u_{n}-u|^{p-2}(u_{n}-u)-|u_{n}|^{p-2}u_{n}+|u|^{p-2}u\| _{X^{-1}} \leq c\Vert\varepsilon_{n}\Vert_{L^{\frac{p}{p-1}}}= o(1). \] $(ii)$ Since \begin{equation} J'(u_{n}) =-\Delta u_{n}+u_{n}-|u_{n}|^{p-2} u_{n}= o(1)\quad\mbox{in }X(\Omega) \label{2-1} \end{equation} and $u_{n}\rightharpoonup u$, then by Lemma \ref{p4}, we have $J'(u)=0$, or \begin{equation} -\Delta u+u-|u|^{p-2}u=0.\label{2-2} \end{equation} Now by part $(i)$, (\ref{2-1}), and (\ref{2-2}), \begin{align*} J'(\varphi_{n}) & =-\Delta\varphi_{n}+\varphi_{n}-|\varphi_{n} |^{p-2}\varphi_{n}\\ & =-\Delta(u_{n}-u) +(u_{n}-u) -|u_{n} -u|^{p-2}(u_{n}-u) \\ & =(-\Delta u_{n}+u_{n}-|u_{n}|^{p-2}u_{n}) -(-\Delta u+u-|u|^{p-2}u) \\ & -(|u_{n}-u|^{p-2}(u_{n}-u) -|u_{n}|^{p-2} u_{n}+|u|^{p-2}u) \\ & = o(1). \end{align*} $(iii)$ Since $u_{n}\rightharpoonup u$ weakly in $X(\Omega) $ and $\{ u_{n}\} $ is a (PS)$_{\beta}$-sequence, by Lemma \ref{p4}, \ref{p7} and Theorem \ref{p21} below, a subsequence $\{ u_{n}\} $ exists such that $a(\varphi _{n}) =a(u_{n}) -a(u) + o(1) $ and $b(\varphi_{n}) =b(u_{n}) -b(u) + o(1) $. Thus, $J(\varphi_{n}) =J(u_{n}) -J(u) + o(1) =\beta-J(u)+ o(1)$. Therefore, by part $(ii)$, $\{ \varphi_{n}\} $ is a (PS)$_{(\beta-J(u)) }$-sequence. \end{proof} Define the concentration function of $|u_{n}|^{2}$ in $\mathbb{R}^{N}$ by \[ Q_{n}(t)=\sup_{z\in\mathbb{R}^{N}}\int_{z+B^{N}(0;t)}|u_{n}|^{2}. \] Then we have the following concentration lemma. \begin{lemma} \label{p9} Let $\{ u_{n}\} $ be bounded in $H^{1}(\mathbb{R}^{N}) $ and for some $t_{0}>0$, let $Q_{n}(t_{0})= o(1)$. Then\newline $(i)$ $u_{n}= o(1)$ strongly in $L^{q}(\mathbb{R} ^{N}) $ for $21$ as $r\to q$, we may choose $r$ satisfying $21$. Recall that \[ \Vert\{a_{n}\}\Vert_{\ell^{s}}=(\overset{\infty}{\underset{n=1}{\sum} }|a_{n}|^{s}) ^{1/s}\leq\overset{\infty}{\underset{n=1}{\sum} }|a_{n}|=\Vert\{a_{n}\}\Vert_{\ell^{1}},\quad \ell^{1}\subset\ell^{2}\subset\dots\subset\ell^{\infty}. \] Thus, \begin{align*} \sum_{i=1}^{\infty}\Big({\int_{P_{i}}} |\nabla u_{n}|^{2}+|u_{n}|^{2}\Big) ^{rt/2} & \leq\Big(\sum_{i=1}^{\infty} {\int_{P_{i}}} (|\nabla u_{n}|^{2}+|u_{n}|^{2}) \Big) ^{s}\\ & =\Big({\int_{\mathbb{R}^{N}}} (|\nabla u_{n}|^{2}+|u_{n}|^{2}) \Big) ^{s}\\ & =\Vert u_{n}\Vert_{H^{1}(\mathbb{R}^{N})}^{2s}\leq c\quad \text{for }n=1,2,\dots. \end{align*} Therefore, \[ {\int_{\mathbb{R}^{N}}} |u_{n}|^{q}\leq c(Q_{n}(t_{0}) )^{(1-t)}, \quad\mbox{or}\quad \int_{\mathbb{R}^{N}}|u_{n}|^{q}= o(1)\quad\text{as }n\to\infty. \] $(ii)$ In addition, if $u_{n}$ satisfies \begin{equation} -\Delta u_{n}+u_{n}-|u_{n}|^{p-2}u_{n}= o(1)\quad\mbox{in }H^{-1}(\mathbb{R} ^{N}),\label{2-3} \end{equation} then $\{u_{n}\}$ is bounded. Multiply Equation (\ref{2-3}) by $u_{n}$ and integrate it to obtain \[ a(u_{n})=b(u_{n})+ o(1). \] By part $(i)$, $b(u_{n})= o(1)$. Thus, $a(u_{n})= o(1)$, or \[ \Vert u_{n}\Vert_{H^{1}}= o(1)\quad\text{strongly in }\,H^{1}(\mathbb{R} ^{N}) . \] \end{proof} \begin{lemma} \label{p10} Let $\{u_{n}\}$ be bounded in $H_{0}^{1}(\mathbf{A}^{r}) $ and for some $t_{0}>0$, \[ Q_{n}^{r}(t_{0})=\sup_{y\in\mathbb{R}}\int_{(0,y) +\mathbf{A} _{-t_{0},t_{0}}^{r}}|u_{n}|^{2}= o(1). \] Then\newline $(i)$ $u_{n}= o(1)$ strongly in $L^{q}(\mathbf{A} ^{r}) $ for $20$, there is a $\delta>0$ such that if $v\in H_{0}^{1}(\Omega) $ solves \eqref{E1} in $\Omega$ satisfying $\Vert v\Vert_{H^{1}}\leq c$ and $\Vert v\Vert_{L^{2}}\leq\delta$, then $v\equiv0$. \end{lemma} \begin{proof} For $00$.\newline $(i)$ Let $\gamma-2\geq0$. Note that $2(1-t_{0})>0$. By (\ref{2-4}), we have \[ 1\leq d\delta^{2(1-t_{0})}\Vert v\Vert_{H^{1}}^{\gamma-2}\leq dc^{\gamma -2}\delta^{2(1-t_{0})}. \] Let $\delta_{1}>0$ satisfy $dc^{\gamma-2}\delta_{1}^{2(1-t_{0})}<1$. If $\delta\leq\delta_{1}$, then \[ 1\leq dc^{\gamma-2}\delta^{2(1-t_{0})}\leq dc^{\gamma-2}\delta_{1} ^{2(1-t_{0})}<1, \] which is a contradiction.\newline $(ii)$ Let $\gamma-2<0$. By (\ref{2-4}), we have \[ \Vert v\Vert_{H^{1}}\leq\delta^{\frac{2(1-t_{0})}{2-\gamma}}d ^{\frac{1}{2-\gamma}}, \] since \[ \Vert v\Vert_{H^{1}}^{2}= {\int_{\Omega}} |v|^{p}\leq c_{1}\Vert v\Vert_{H^{1}}^{p}, \quad\mbox{or}\quad 1\leq c_{1}\Vert v\Vert_{H^{1}}^{p-2}. \] Thus, we have \[ 1\leq c_{1}\Vert v\Vert_{H^{1}}^{p-2}\leq c_{2}\delta^{\frac{2(1-t_{0} )(p-2)}{2-\gamma}}, \] where $c_{2}=c_{1}d^{\frac{p-2}{2-\gamma}}>0$. Note that $\frac{2(1-t_{0} )(p-2)}{2-\gamma}>0$. Let $\delta_{2}>0$ such that \[ c_{2}\delta_{2}^{\frac{2(1-t_{0})(p-2)}{2-\gamma}}<1. \] If $\delta\leq\delta_{2}$, then $1\leq c_{2}\delta^{\frac{2(1-t_{0} )(p-2)}{2-\gamma}}<1$, which is a contradiction.\newline Take $\delta_{0} =\min\{ \delta_{1}\text{, }\delta_{2}\} $, if $\delta\leq \delta_{0}$, from parts $(i)$ and $(ii)$, and we obtain $\Vert v\Vert_{H^{1} }=0$ or $v=0$. \end{proof} Let \[ \tilde{u}(z)=\begin{cases} u(z) & \text{for } z\in\Omega;\\ 0 & \text{for } z\in\mathbb{R}^{N}\backslash \Omega. \end{cases} \] Then we have the following characterization of a function in $W_{0}^{1,p}(\Omega)$. \begin{lemma} \label{p12} Let $\Omega$ be a $C^{0,1}$ domain in $\mathbb{R}^{N}$ and $u\in L^{p}(\Omega)$ with $10$ such that \[ | \int_{\Omega}u\frac{\partial\varphi}{\partial x_{i}}| \leq c\| \varphi\| _{L^{p}},\quad\text{for each }\varphi\in C_{c} ^{1}(\mathbb{R}^{N})\text{, }i=1,2,\dots,N; \] $(iii)$ $\tilde{u}\in W_{0}^{1,p}(\mathbb{R}^{N})$ and $\frac{\partial\widetilde{u}}{\partial z_{i}} =\frac{\widetilde{\partial u}}{\partial z_{i}}$. \end{lemma} For the proof of this lemma, see Br\'{e}zis \cite[Proposition IX.18]{B}, Gilbarg-Trudinger \cite[Theorem 7.25]{GT}, and Grisvard \cite[p26]{G}. We recall the classical compactness theorems. The Lebesgue dominated convergence theorem is a well-known compactness theorem. \begin{theorem}[Lebesgue Dominated Convergence Theorem] \label{p13} Suppose $\Omega$ is a domain in $\mathbb{R}^{N}$, $\{ u_{n}\} _{n=1}^{\infty}$ and $u$ are measurable functions in $\Omega$ such that $u_{n}\to u$ a.e. in $\Omega$. If $\varphi\in L^{1}(\Omega)$ exists such that for each $n$ \[ | u_{n}| \leq\varphi\quad\text{a.e. in }\Omega, \] then $u_{n}\to u$ in $L^{1}(\Omega)$. \end{theorem} The converse of the Lebesgue dominated convergence theorem fails. \begin{example} \label{p14} \rm For $n=1,2,\dots$, let $u_{n}:\mathbb{R\to R}$ be defined by \[ u_{n}(z)=\begin{cases} 0 & \text{for }z\leq n;\\ 2 & \text{for }z=n+1/2n;\\ 0 & \text{for }z\geq n+1/n;\\ &\text{linear otherwise.} \end{cases} \] \begin{figure}[htb] \begin{center} \includegraphics[width=0.7\textwidth]{fig03} \end{center} % \centering \resizebox{4.5in}{!}{\includegraphics{./fig03.eps}} \caption{Counter example 1.} \label{fig:fig03} \end{figure} We have \[ {\int_{\mathbb{R}}} u_{n}(z)dz = \frac{1}{n}<\infty\quad \quad\text{for each }n\in\mathbb{N}. \] Hence, $u_{n}\to 0$ a.e. in $\mathbb{R}$ and strongly in $L^{1} (\mathbb{R}) $. Let $\varphi:\mathbb{R\to R}$ satisfy $|u_{n}| \leq\varphi$ a.e. in $\mathbb{R}$ for each $n\in\mathbb{N}$. Then $\infty=\underset{n=1}{\overset{\infty}{\sum}}\frac{1}{n}= {\int_{\mathbb{R}}} \underset{n=1}{\overset{\infty}{\sum}}u_{n}\leq {\int_{\mathbb{R}}}\varphi$. Consequently, $\varphi\notin L^{1}(\mathbb{R})$. \end{example} However, the generalized Lebesgue dominated convergence theorem is a necessary and sufficient result for $L^{1}$ convergence. \begin{theorem}[Generalized Lebesgue Dominated Convergence Theorem:] \label{p15} Suppose $\Omega$ is a domain in $\mathbb{R}^{N}$, $\{ u_{n}\} _{n=1}^{\infty}$ and $u$ are measurable functions in $\Omega$ such that $u_{n}\to u$ a.e. in $\Omega$. Then $u_{n}\to u$ in $L^{1}(\Omega)$ if and only if $\{ \varphi_{n}\} _{n=1}^{\infty },\varphi\in L^{1}(\Omega)$ exist such that $\varphi_{n}\to\varphi$ a.e. in $\Omega$, $| u_{n}| \leq\varphi_{n}$ a.e. in $\Omega$ for each $n$, and $\varphi_{n}\to\varphi$ in $L^{1}(\Omega)$. \end{theorem} \begin{proof} ($\Longrightarrow$) Suppose that $u_{n}\to u$ in $L^{1}(\Omega)$, take $\varphi_{n}=| u_{n}| $ and $\varphi=| u| $, then $\varphi_{n}\to\varphi$ in $L^{1}(\Omega)$. \newline ($\Longleftarrow$) Suppose that a sequence of measurable functions $\{ \varphi_{n}\} _{n=1}^{\infty}$ and $\varphi$ in $\Omega$ exist such that $\varphi_{n}\in L^{1}(\Omega)$, $\varphi_{n}\to\varphi$ a.e. in $\Omega$, $| u_{n}| \leq\varphi_{n}$ a.e. in $\Omega$ for each $n$, and $\varphi _{n}\to\varphi$ in $L^{1}(\Omega)$. Applying the Fatou lemma, we have \[ \int_{\Omega}\liminf_{n\to\infty}(\varphi_{n}-u_{n})\leq \liminf_{n\to\infty}\int_{\Omega}(\varphi_{n}-u_{n}), \] or \[ \int_{\Omega}u\geq\limsup_{n\to\infty}\int_{\Omega}u_{n}. \] Applying the Fatou lemma again, we have \[ \int_{\Omega}\liminf_{n\to\infty}(\varphi_{n}+u_{n})\leq \liminf_{n\to\infty}\int_{\Omega}(\varphi_{n}+u_{n}), \] or \[ \int_{\Omega}u\leq\liminf_{n\to\infty}\int_{\Omega}u_{n}. \] Thus, \[ \int_{\Omega}u=\lim_{n\to\infty}\int_{\Omega}u_{n}. \] \end{proof} Another necessary and sufficient result for $L^{1}$ convergence is the Vitali convergence theorem. \begin{theorem}[Vitali Convergence Theorem for $L^{1}(\Omega)$] \label{p16} Suppose $\Omega$ is a domain in $\mathbb{R}^{N}$, $\{ u_{n}\} _{n=1}^{\infty}$ in $L^{1}(\Omega) $, and $u\in L^{1}(\Omega) $. Then $\| u_{n}-u\| _{L^{1}}\to0$ if the following three conditions hold:\newline $(i)$ $u_{n}\to u$ a.e in $\Omega$; \newline $(ii)$ (Uniformly integrable) For each $\varepsilon>0$, a measurable set $E\subset\Omega$ exists such that $| E| <\infty$ and \[ \int_{E^{c}}| u_{n}| d\mu<\varepsilon \] for each $n\in\mathbb{N}$, where $E^{c}=\Omega\backslash E$; \newline $(iii)$ (Uniformly continuous) For each $\varepsilon>0$, $\delta>0$ exists such that $|E|<\delta$ implies \[ \int_{E}| u_{n}| d\mu<\varepsilon\quad \text{for each }n\in \mathbb{N}. \] Conversely, if $\| u_{n}-u\| _{L^{1}}\to0$, then conditions $(ii)$ and $(iii)$ hold and there is a subsequence $\{u_{n}\}$ such that $(i)$ holds. Furthermore, if $| \Omega| <\infty$, then we can drop condition $(ii)$. \end{theorem} \begin{proof} Assume the three conditions hold. Choose $\varepsilon>0$ and let $\delta>0$ be the corresponding number given by condition $(iii)$. Condition $(ii)$ provides a measurable set $E\subset\Omega$ with $| E| <\infty$ such that \[ \int_{E^{c}}| u_{n}| d\mu<\varepsilon \] for all positive integers $n$. Since $| E| <\infty$, we can apply the Egorov theorem to obtain a measurable set $B\subset E$ with $| E \backslash B| <\delta$ such that $u_{n}$ converges uniformly to $u$ on $B$. Now write \[ \int_{\Omega}| u_{n}-u| d\mu =\int_{B}| u_{n}-u|d\mu +\int_{E\backslash B}| u_{n}-u| d\mu+\int_{E^{c}}| u_{n}-u| d\mu. \] Since $u_{n}\to u$ uniformly in $B$, the first integral on the right can be made arbitrarily small for large $n$. The second and third integrals will be estimated with the help of the inequality \[ | u_{n}-u| \leq| u_{n}| +| u| . \] From condition $(iii)$, we have $\int_{E\backslash B}| u_{n}| d\mu<\varepsilon$ for all $n\in\mathbb{N}$ and the Fatou Lemma shows that $\int_{E\backslash B}| u| d\mu\leq\varepsilon$ as well. The third integral can be handled in a similar way using condition $(ii)$. Thus, it follows that $\| u_{n}-u\| _{L^{1}}\to0$. Now suppose $\| u_{n}-u\| _{L^{1}}\to0$. Then for each $\varepsilon>0$, a positive integer $n_{0}$ exists such that $\| u_{n}-u\| _{L^{1}}<\varepsilon/2$ for $n>n_{0}$, and measurable sets $A$ and $B$ of finite measure exist such that \[ \int_{A^{c}}| u| d\mu<\varepsilon/2\quad\text{and }\int_{B^{c} }| u_{n}| d\mu<\varepsilon \quad\text{for }n=1,2,\dots ,n_{0}. \] Minkowski's inequality implies that \[ \| u_{n}\| _{L^{1}(A^{c})}\leq\| u_{n}-u\| _{L^{1}(A^{c})}+\| u\| _{L^{1}(A^{c})}<\varepsilon\quad \text{for }n>n_{0.} \] Then let $E=A\cup B$ to obtain the necessity of condition $(ii)$. Similar reasoning establishes the necessity of condition $(iii)$. Convergence in $L^{1}$ implies convergence in measure. Hence, condition $(i)$ holds for a subsequence. \end{proof} There is a bounded sequence $\{u_{n}\}$ in $L^{1}(\mathbb{R})$ that violates Theorem \ref{p16} condition $(ii)$. \begin{example} \label{p17} \rm For $n=1,2,\dots$, let $u_{n}:\mathbb{R\to R}$ be defined by \[ u_{n}(z)=\begin{cases} 0 & \text{for }z\leq n;\\ 2 & \text{for }z=n+1/2;\\ 0 & \text{for }z\geq n+1;\\ &\text{linear otherwise,} \end{cases} \] \begin{figure}[htb] \begin{center} \includegraphics[width=0.5\textwidth]{fig04} \end{center} % \centering \resizebox{2.5in}{!}{\includegraphics{./fig04.eps}} \caption{ counter example violating Theorem \ref{p16} condition $(ii)$.} \label{fig:fig04} \end{figure} then ${\int_{\mathbb{R}}}u_{n}(z)dz=1$ for each $n\in\mathbb{N}$. Clearly, $\{u_{n}\}$ violates Theorem \ref{p16} $(ii)$. \end{example} There is a bounded sequence $\{u_{n}\}$ in $L^{1}(\mathbb{R})$ that violates Theorem \ref{p16} condition $(iii)$. \begin{example}\label{p18} \rm For $n=1,2,\dots$, let $u_{n}:\mathbb{R\to R}$ be defined by \[ u_{n}(z)=\begin{cases} 0 & \text{for }z\leq n;\\ 2n & \text{for }z=n+1/2n;\\ 0 & \text{for }z\geq n+1/n;\\ &\text{linear therwise.} \end{cases} \] \begin{figure}[htb] \begin{center} \includegraphics[width=0.5\textwidth]{fig05} \end{center} % \centering \resizebox{2.5in}{!}{\includegraphics{./fig05.eps}} \caption{ counter example violating Theorem \ref{p16} condition $(iii)$.} \label{fig:fig05} \end{figure} Then \[ {\int_{\mathbb{R}}} u_{n}(z)dz=1\quad\text{for each }n\in\mathbb{N}. \] Clearly, $\{u_{n}\}$ violates Theorem \ref{p16} condition $(iii)$. \end{example} \begin{lemma} \label{p19} In the Vitali convergence theorem \ref{p16} condition $(ii)$, the set $E$ with $|E| <\infty$ can be replaced by the condition that $E$ is bounded. \end{lemma} \begin{proof} Let $E_{n}=E\cap B^{N}(0;n)$ for $n=1,2,\dots$. Then $E_{1}\subset E_{2}\subset\dots\nearrow E$. Thus $| E_{1}| \leq| E_{2}| \leq\dots\nearrow| E| $. For $\delta>0$ as in Theorem \ref{p16} condition $(iii)$, there is an $E_{N}$ such that $| E\backslash E_{N}| <\delta$. Now \[ \int_{E_{N}^{c}}| u_{n}| dz=\int_{E^{c}}| u_{n}| dz+\int_{E\backslash E_{N}}| u_{n}| dz<2\varepsilon \] for each $n\in\mathbb{N}$. \end{proof} \begin{lemma} \label{p20} Let $\Omega$ be a domain in $\mathbb{R}^{N}$, $1\leq r0$, then $W_{0} ^{m,p}(\Omega)\hookrightarrow L^{q}(\Omega)$, where $q\in[p,p^{*}]$, $\frac {1}{p^{*}}=\frac{1}{p}-\frac{m}{N}$; \newline $(ii)$ If $\frac{1}{p}-\frac{m} {N}=0$, then $W_{0}^{m,p}(\Omega)\hookrightarrow L^{q}(\Omega)$, where $q\in[p,\infty)$; \newline $(iii)$ If $\frac{1}{p}-\frac{m}{N}<0$, then $W_{0}^{m,p}(\Omega)\hookrightarrow L^{\infty}(\Omega)$.\newline Moreover, if $m-\frac{N}{p}>0$ is not an integer, let $k=\left[ m-\frac{N}{p}\right] $ and $\theta=m-\frac{N}{p}-k$ $(0<\theta<1)$, then we have for $u\in W_{0}^{m,p}(\Omega)$ \begin{gather*} \| D^{\beta}u\| _{L^{\infty}}\leq c\| u\| _{W^{m,p}} \quad \text{for }|\beta|\leq k\\ | u(x)-u(y)| \leq c\| u\| _{W^{m,p}}| x-y| ^{\theta}\quad \quad\text{a.e. for }x,y\in\Omega. \end{gather*} In particular, $W_{0}^{m,p}(\Omega)\hookrightarrow C^{k,\theta}(\overline {\Omega})$. \end{theorem} For the proof ot the theorem above, see Gilbarg-Trudinger \cite[p.164]{GT}. \begin{definition} \label{p22} \rm $\Omega$ satisfies a uniform interior cone condition if a fixed cone $K_{\Omega}$ exists such that each $x\in\partial\Omega$ is the vertex of a cone $K_{\Omega}(x)\subset\overline{\Omega}$ and congruent to $K_{\Omega}$. \end{definition} \begin{theorem}[Sobolev Embedding Theorem in $W^{m,p}(\Omega)$]\label{p23} Let $\Omega$ satisfy a uniform interior cone condition, $m\in\mathbb{N}$ and $1\leq p<\infty$. Then we have the following continuous injections.\newline $(i)$ If $\frac{1}{p}-\frac{m}{N}>0$, then $W^{m,p}(\Omega)\hookrightarrow L^{q} (\Omega)$, where $q\in[p,p^{*}]$ and $\frac{1}{p^{*}}=\frac{1}{p}-\frac{m}{N}$; \newline $(ii)$ If $\frac{1}{p}-\frac{m}{N}=0$, then $W^{m,p} (\Omega)\hookrightarrow L^{q}(\Omega)$, where $q\in[p,\infty)$; \newline $(iii)$ If $\frac{1}{p}-\frac{m}{N}<0$, then $W^{m,p}(\Omega)\hookrightarrow L^{\infty}(\Omega)$.\newline Moreover, if $m-\frac{N}{p}>0$ is not an integer, let \[ k=\big[ m-\frac{N}{p}\big] \quad \text{and}\quad \theta=m-\frac{N}{p}-k\quad (0<\theta<1), \] then we have for $u\in W^{m,p}(\Omega)$, \begin{gather*} \| D^{\beta}u\| _{L^{\infty}}\leq c\| u\| _{W^{m,p} }\quad\text{for }\beta\quad\text{with }|\beta|\leq k\\ | D^{\beta}u(x)-D^{\beta}u(y)| \leq c\| u\| _{W^{m,p}}| x-y| ^{\theta}\quad\text{a.e. for }x,y\in\Omega\quad\text{and }|\beta|=k. \end{gather*} In particular, $W^{m,p}(\Omega)\hookrightarrow C^{k,\theta}(\overline{\Omega })$. \end{theorem} For the proof of the theorem above, see Br\'{e}zis \cite[Cor. IX.13]{B} and Gilbarg-Trudinger \cite[Theorem 7.26]{GT}. \begin{theorem}[Rellich-Kondrakov Theorem in $W_{0}^{m,p}(\Omega)$] \label{p24} Let $\Omega$ be a bounded domain, $m\in\mathbb{N}$ and $1\leq p<\infty$. Then we have the following compact injections.\newline $(i)$ If $\frac{1}{p}-\frac{m}{N}>0$, then $W_{0}^{m,p}(\Omega)\hookrightarrow L^{q}(\Omega)$, where $q\in[1,p^{*} )$, $\frac{1}{p^{*}}=\frac{1}{p}-\frac{m}{N}$; \newline $(ii)$ If $\frac{1} {p}-\frac{m}{N}=0$, then $W_{0}^{m,p}(\Omega)\hookrightarrow L^{q}(\Omega)$, where $q\in[1,\infty)$; \newline $(iii)$ If $\frac{1}{p}-\frac{m}{N}<0$, then $W_{0}^{m,p}(\Omega)\hookrightarrow C^{k}(\overline{\Omega})$, where $m-\frac{N}{p}>0$ is not an integer and $k=\left[ m-\frac{N}{p}\right] $. \end{theorem} For the proof of the aboved theroem, see Gilbarg-Trudinger \cite[Theorem 7.22]{GT}. \begin{theorem}[Rellich-Kondrakov Theorem in $W^{m,p}(\Omega)$] \label{p25} Let $\Omega$ be a bounded $C^{0,1}$ domain in $\mathbb{R}^{N}$, $m\in\mathbb{N}$ and $1\leq p<\infty$. Then we have the following compact injections.\newline$(i)$ If $\frac{1}{p}-\frac{m}{N}>0$, then $W^{m,p}(\Omega)\hookrightarrow L^{q} (\Omega)$, where $q\in[1,p^{*})$, $\frac{1}{p^{*}}=\frac{1}{p}-\frac{m}{N} $; \newline$(ii)$ If $\frac{1}{p}-\frac{m}{N}=0$, then $W^{m,p}(\Omega )\hookrightarrow L^{q}(\Omega)$, where $q\in[1,\infty)$; \newline$(iii)$ If $\frac{1}{p}-\frac{m}{N}<0$, then $W^{m,p}(\Omega)\hookrightarrow C^{k,\beta }(\overline{\Omega})$, where $m-\frac{N}{p}>0$ is not an integer, $0<\beta<\theta$, $k=\left[ m-\frac{N}{p}\right] $, and $\theta=m-\frac {N}{p}-k$ $(0<\theta<1)$. \end{theorem} For the proof of the above theorem, see Br\'{e}zis \cite[p. 169]{B} and Gilbarg-Trudinger \cite[Theorem 7.26]{GT}. For the Sobolev space $X(\Omega)$, we can drop condition $(iii)$ of the Vitali convergence theorem \ref{p16} through the interpolation results. \begin{theorem}[Rellich-Kondrakov Theorem]\label{p26} Let $\Omega$ be a domain in $\mathbb{R}^{N}$ of finite measure. Then the embedding $X(\Omega )\hookrightarrow L^{p}(\Omega) $ is compact. \end{theorem} \begin{proof} Let $\{ u_{n}\} $ be a bounded sequence in $X(\Omega)$, then by Lemma \ref{p4}, a subsequence $\{ u_{n}\} $ and $u\in X(\Omega)$ exist such that $u_{n}\to u$ a.e. in $\Omega$. By the Egorov theorem, for $\varepsilon>0$, a closed subset $F$ in $\mathbb{R}^{N}$ exists such that $F\subset\Omega$, $| \Omega\backslash F| <\varepsilon$, and $u_{n}\to u$ uniformly in $F$. Thus, \[ \int_{F}| u_{n}-u| ^{p}= o(1)\quad \text{as }n\to\infty. \] For $N>2$, we have \begin{align*} {\int_{\Omega\backslash F}} | u_{n}-u| ^{p} & \leq\Big( {\int_{\Omega\backslash F}}1\Big) ^{1/r} \Big({\int_{\Omega\backslash F}}| u_{n}-u| ^{ps}\Big) ^{1/s}\\ & \leq| \Omega\backslash F| ^{1/r}\Big( {\int_{\Omega}} | u_{n}-u| ^{ps}\Big) ^{1/s}\\ & \leq c\| u_{n}-u\| _{H^{1}}^{p}| \Omega\backslash F| ^{1/r} 1$ to obtain the above inequality. Hence, $u_{n}\to u$ strongly in $L^{p}(\Omega) $. \end{proof} \begin{theorem}[Vitali Convergence Theorem for $X(\Omega)$] \label{p27} $(i)$ Let $\Omega$ be a domain in $\mathbb{R}^{N}$ of finite measure. Then the embedding $X(\Omega)\hookrightarrow L^{p}( \Omega) $ is compact;\newline $(ii) $ Let $\Omega$ be a domain in $\mathbb{R}^{N}$ and let $\{ u_{n}\} _{n=1}^{\infty}$ be a sequence in $X(\Omega)$. Suppose that a constant $c>0$ exists such that $\| u_{n}\| _{H^{1}}\leq c$ for each $n$ and $u_{n}\to u$ a.e. in $\Omega$. Then for each $\varepsilon>0$, a measurable set $E\subset\Omega$ exists such that $| E| <\infty$ and $\int _{E^{c}}| u_{n}| ^{p}dz<\varepsilon$\ for each $n\in\mathbb{N} \ $if and only if $\| u_{n}-u\| _{_{L^{p}(\Omega)}}= o(1)$. \end{theorem} \begin{proof} Part $(i)$ follows from Willem \cite{Wi}. $(ii)$ By the Fatou lemma, $\int_{E^{c}}| u| ^{p}dz\leq\varepsilon$. Since $| E| <\infty$ and $\| u_{n}\| _{H^{1}}\leq c$, by $(i) $, there is a subsequence $\{ u_{n}\} _{n=1}^{\infty}$ satisfying \[ \int_{E}| u_{n}-u| ^{p}dz= o(1). \] Therefore, \[ \int_{\Omega}| u_{n}-u| ^{p}dz=\int_{E\cap\Omega}| u_{n}-u| ^{p}dz+\int_{E^{c}\cap\Omega}| u_{n}-u| ^{p}dz= o(1). \] Now suppose $\| u_{n}-u\| _{L^{p}(\Omega)}= o(1)$. Then for each $\varepsilon>0$, a positive integer $n_{0}$ exists such that $\| u_{n}-u\| _{L^{p}(\Omega)}<\frac{\varepsilon^{1/p}}{2}$ for $n>n_{0}$, and measurable sets $A$ and $B$ of finite measure exist such that \[ \int_{A^{c}}| u| ^{p}dz<\frac{\varepsilon}{2^{p}}\quad\text{and}\quad \int_{B^{c}}| u_{n}| ^{p}dz<\varepsilon\quad \text{for }n=1,2,\dots ,n_{0}. \] The Minkowski inequality implies \[ \| u_{n}\| _{L^{p}(A^{c})}\leq\| u_{n}-u\| _{L^{p}(A^{c})}+\| u\| _{L^{p}(A^{c})}<\varepsilon^{1/p}\quad\text{for }n>n_{0}. \] Then let $E=A\cup B$ to obtain the conclusion. \end{proof} Let $L_{w}^{p}(\mathbb{R}^{N})=\{ u\in L_{\rm loc}^{p}(\mathbb{R}^{N}): \int_{\mathbb{R}^{N}}| u(z) | ^{p}w(z) dz<\infty\} $ be a weighted Lebesgue space, where the weight $w$ is nonnegative with \[ \| u\| _{L_{w}^{p}(\mathbb{R}^{N})}^{p}=\int_{\mathbb{R}^{N} }| u(z) | ^{p}w(z) dz. \] We denote by $Q(x,l) $ the cube of the form \[ Q(x,l) =\{ y\in\mathbb{R}^{N}: | y_{j}-x_{j}| 2$. Suppose that $w\in$ $L_{w}^{\frac{p+\delta}{\delta}}(\mathbb{R}^{N})$, with $2\leq p0$, and \begin{equation} \lim_{| x| \to\infty}\int_{Q(x,l) }w( z) ^{^{\frac{p+\delta}{\delta}}}dz=0\label{A1} \end{equation} for some $l>0$. Then $H^{1}(\mathbb{R}^{N})$ is compactly embedded in $L_{w}^{p}(\mathbb{R}^{N})$; \newline$(ii) $ Let $N=2$ and suppose that $w\in$ $L_{w}^{s}(\mathbb{R}^{N})$ for some $s>1$ and \begin{equation} \lim_{|x| \to\infty}\int_{Q(x,l) }w(z) ^{^{s}}dz=0\label{A2} \end{equation} for some $l>0$. Then $H^{1}(\mathbb{R}^{N})$ is compactly embedded in $L_{w}^{p}(\mathbb{R}^{N})$ for every $p\geq2$; \newline$(iii) $ Let $N=1$ and suppose that $w\in$ $L_{\rm loc}^{1}(\mathbb{R}^{N})$ and \begin{equation} \lim_{| x| \to\infty}\int_{Q(x,l) }w( z) dz=0\label{A3} \end{equation} for some $l>0$. Then $H^{1}(\mathbb{R}^{N})$ is compact embedded in $L_{w} ^{p}(\mathbb{R}^{N})$ for every $p\geq2$. \end{theorem} \begin{proof} $(i) $ It suffices to show that for every $\varepsilon>0$, a $R>0$ exists such that \begin{equation} \| u-u\chi_{Q(0,R) }\| _{L_{w}^{p}(\mathbb{R}^{N} )}<\varepsilon\label{A4} \end{equation} for each $u\in H^{1}(\mathbb{R}^{N})$ such that $\| u\| _{H^{1}(\mathbb{R}^{N})}\leq1$, where $\chi_{Q}$ is the characteristic function of the cube. Indeed, let $\{ u_{n}\} $ be a bounded sequence in $H^{1}(\mathbb{R}^{N})$. We assume that $\| u_{n}\| _{H^{1}(\mathbb{R}^{N})}\leq1$ for all $n\in\mathbb{N}$. Consequently, a subsequence $\{ u_{n}\} $ and a $u\in H^{1}(\mathbb{R}^{N})$ exist such that $u_{n}\rightharpoonup u$ in $H^{1}(\mathbb{R}^{N})$ and $u_{n}\to u$ in $L^{p}(Q(0,R) ) $. On the other hand, by \eqref{A4}, we have \[ \| u_{n}-u\| _{L_{w}^{p}(\mathbb{R}^{N}\backslash Q( 0,R) )}\leq\| u_{n}\| _{L_{w}^{p}(\mathbb{R}^{N}\backslash Q(0,R) )}+\| u\| _{L_{w}^{p}(\mathbb{R} ^{N}\backslash Q(0,R) )}\leq2\varepsilon. \] Combining this with the previous observation, it is easy to conclude that $u_{n}\to u$ in $L_{w}^{p}(\mathbb{R}^{N}) $. To show \eqref{A4}, we cover $\mathbb{R}^{N}$ with cubes $Q(\hat{z},1) $, $\hat{z}\in\mathbb{Z}^{N}$. We may assume that $(i) $ holds with $l=1$. For $\eta>0$, we use $( \text{\ref{A1}}) $ to find a positive constant $n_{0}$ such that $\int_{Q}w(z) ^{\frac{p+\delta}{\delta}}dz<\eta$ for each $Q=Q(\hat{z},1) $ outside $Q(0,n_{0}) $. By the Sobolev embedding theorem, for any $u\in H^{1}(\mathbb{R}^{N}) $, a constant $c>0$ exists such that \[ \| u\| _{L^{p}(Q) }\leq c\| u\| _{H^{1}(Q) }\quad\text{for all }2\leq p<2^{*}. \] Thus, by the H\"{o}lder inequality, we have \[ \int_{Q}| u| ^{p}wdz\leq\Big(\int_{Q}w^{\frac{p+\delta} {\delta}}dz\Big) ^{\frac{\delta}{p+\delta}} \Big(\int_{Q}|u| ^{p+\delta}dz\Big) ^{\frac{p}{p+\delta}} \leq c'\eta^{1/s}\| u\| _{H^{1}(Q) }^{p} \] where $c'=c^{p/(p+\delta)}$. Now, choose $c'\eta^{1/s}<\varepsilon$ and add these inequalities over all $Q(\hat{z},1) $ outside $Q(0,n_{0}) $ to obtain $R=n_{0}$. \newline $(ii)$ and $(iii)$ are similar to $(i)$. \end{proof} We define $H_{r}^{1}(\Omega) =\{ u\in H_{0}^{1}(\Omega): u \quad\text{is radially symmetric}\} $. \begin{lemma}\label{p28} For $N\geq2$, every $u\in H_{r}^{1}(\mathbb{R}^{N})$ is equal to a continuous function $U$ a.e. in $\mathbb{R}^{N}\backslash\{0\}$ such that for $z\neq0$ \[ |U(z)|\leq(\frac{2}{\omega_{N}}) ^{1/2}|z|^{\frac{1-N}{2}} \Big(\int_{|t|\geq|z|}|u(t)|^{2}dt\Big) ^{1/4} \Big(\int_{|y|\geq|z|}|\nabla u(t)|^{2}dt\Big) ^{1/4}, \] where $\omega_{N}$ is the area of the unit ball in $\mathbb{R}^{N}$. \end{lemma} \begin{proof} Let $\varphi\in C_{c}^{\infty}(\mathbb{R}^{N})$ be a radially symmetric function. Then for $0\leq r<\infty$, \begin{align*} r^{N-1}\varphi(r)^{2} & =\int_{0}^{r}(s^{N-1}\varphi(s)^{2}) 'ds\\ & =(N-1)\int_{0}^{r}s^{N-2}\varphi(s)^{2}ds+2\int_{0}^{r}s^{N-1} \varphi(s)\varphi'(s)ds. \end{align*} Thus, \[ 0=(N-1)\int_{0}^{\infty}s^{N-2}\varphi(s)^{2}ds+2\int_{0}^{\infty} s^{N-1}\varphi(s)\varphi'(s)ds. \] Consequently, \begin{align*} r^{N-1}\varphi(r)^{2} & \leq(N-1){\int_{0}^{\infty}} s^{N-2}\varphi(s)^{2}ds+2{\int_{0}^{r}} s^{N-1}\varphi(s)\varphi'(s)ds\\ & =-2 {\int_{r}^{\infty}} s^{N-1}\varphi(s)\varphi'(s)ds\\ & =(\frac{-2}{\omega_{N}}){\int_{|t|\geq r}} \varphi(t)\varphi'(t)dt\\ & \leq(\frac{2}{\omega_{N}}) \Big({\int_{|t|\geq r}} |\varphi(t)|^{2}dt\Big) ^{1/2}\Big({\int_{|t|\geq r}} |\nabla\varphi(t)|^{2}dt\Big) ^{1/2}. \end{align*} For $u\in H_{r}^{1}(\mathbb{R}^{N})$, take a sequence $\{\varphi_{n}\}$ radially symmetric in $C_{c}^{\infty}(\mathbb{R}^{N})$, such that \[ \varphi_{n}\to u\quad\mbox{in }H^{1}(\mathbb{R}^{N}), \] then there is a subsequence $\{\varphi_{n}(r)\}$ such that \begin{align*} r^{N-1}u(r)^{2} & =\underset{n\to\infty}{\lim}r^{N-1}\varphi _{n}(r)^{2}\\ & \leq\lim_{n\to\infty}(\frac{2}{\omega_{N}}) \Big(\int_{|t|\geq r}|\varphi_{n}(t)|^{2}dt\Big) ^{1/2} \Big(\int_{|t|\geq r}|\nabla\varphi_{n}(t)|^{2}dt\Big) ^{1/2}\\ & \leq(\frac{2}{\omega_{N}}) \Big(\int_{|t|\geq r} |u(t)|^{2}dt\Big) ^{1/2} \Big(\int_{|t|\geq r}|\nabla u(t)|^{2}dt\Big)^{1/2}. \end{align*} Since $u\in H_{r}^{1}(\mathbb{R}^{N})$, it is a function in $H^{1} (\mathbb{R})$, and there is a continuous function $U$ in $\mathbb{R}$ such that $u=U$ a.e. and \[ |U(z)|\leq(\frac{2}{\omega_{N}}) ^{1/2}|z|^{\frac{1-N}{2} }\Big(\int_{|t|\geq|z|}|u(t)|^{2}dt\Big) ^{1/4} \Big(\int_{|t|\geq |z|}|\nabla u(t)|^{2}dt\Big) ^{1/4}. \] \end{proof} Let $\Theta$ be an annulus, say $\Theta=\{ z\in\mathbb{R} ^{N}: 1<| z| \} $ with $N\geq3$. \begin{theorem}[Rellich-Kondrakov Theorem for $H_{r}^{1}(\Theta)$] \label{p29} The embedding \\ $H_{r}^{1}(\Theta)\hookrightarrow L^{p}(\Theta)$ is compact. \end{theorem} \begin{proof} Let $\{ u_{n}\} $ be a bounded sequence in $H_{r}^{1}(\Theta)$. Then a subsequence $\{ u_{n}\} $ exists such that $u_{n} \to u$ a.e. in $\Theta$ and $u_{n}\rightharpoonup u$ weakly in $H_{0}^{1}(\Theta)$. By Lemma \ref{p28}, $\underset{|z|\to\infty} {\lim}u_{n}(z)=0$ uniformly in $n$ and $\underset{s\to0}{\lim} \frac{|s|^{p}}{|s|^{2}+|s|^{2^{*}}}=0$. Thus, for $\varepsilon>0$, there is a $K>0$ such that if $|z|\geq K$, for each $n$, we have \[ | u_{n}(z)| ^{p}\leq\varepsilon(| u_{n}(z)| ^{2}+| u_{n}(z)| ^{2^{*}}) , \] or \[ {\int_{\Theta_{K}^{c}}} | u_{n}| ^{p}\leq c\varepsilon, \] where $\Theta_{K}=\{ z\in\Theta : \ z|0$, then $c>0$ exists such that $\| u_{n}\| _{H^{1}}\geq c$ for each $n$. \end{lemma} \begin{proof} Suppose that a subsequence $\{u_{n}\}$ satisfies $\underset{n\to \infty}{\lim}\| u_{n}\| _{H^{1}}$ $=0$. Then $J(u_{n})= o(1)$, but this contradicts $\beta>0$. Thus, $c>0$ exists such that $\| u_{n}\|_{H^{1}}\geq c$ for each $n$. \end{proof} Let $\Omega$ be an unbounded domain and $\xi_{n}$ as in (\ref{1-1}), then we have the following lemma. \begin{lemma} \label{p32} Let $\{ u_{n}\} $ be a (PS)$_{\beta}$-sequence in $X(\Omega) $ for $J$ such that \[ \int_{\Omega_{n}}|u_{n}|^{p}= o(1), \] where $\Omega_{n}=\Omega\cap B^{N}(0;n)$. Then for any $r\geq1$, we have\newline $(i)$ ${\int_{\Omega}} \xi_{n}^{r}|u_{n}|^{p}= {\int_{\Omega}} |u_{n}|^{p}+ o(1)=\frac{2p}{p-2}\beta+\text{o} (1)$; \newline $(ii)$ ${\int_{\Omega}} \xi_{n}^{r}(|\nabla u_{n}|^{2}+u_{n}^{2})= {\int_{\Omega}} \xi_{n}^{r}|u_{n}|^{p}+ o(1)=\frac{2p}{p-2}\beta + o(1)$; \newline $(iii)$ ${\int_{\Omega}} (\xi_{n}^{r}-1)u_{n}\varphi= o(1)\| \varphi\| _{H^{1}}$ for every $\varphi\in X(\Omega)$; \newline $(iv)$ $|{\int_{\Omega}} (\xi_{n}^{r}-1) | u_{n}| ^{p-2}u_{n}\varphi| = o(1)\| \varphi\| _{H^{1}}$ for every $\varphi\in X(\Omega)$; \newline $(v)$ $|{\int_{\Omega}} (\xi_{n}^{r}-1) \nabla u_{n}\nabla\varphi| = o(1)\| \varphi\| _{H^{1}}$ for every $\varphi\in X(\Omega)$. \end{lemma} \begin{proof} $(i)$ Clearly, we have \[ \int_{\Omega}\xi_{n}^{r}|u_{n}|^{p}=\int_{\Omega}|u_{n} |^{p}+ o(1)=\frac{2p}{p-2}\beta+ o(1). \] $(ii)$ Let $w_{n}=\xi_{n}^{r}u_{n}$. Since $\{w_{n}\}$ is bounded in $X(\Omega)$, we have \begin{align*} o(1) & =\langle J'(u_{n}),w_{n}\rangle \\ & =\int_{\Omega}(\xi_{n}^{r}|\nabla u_{n}|^{2}+r\xi_{n}^{r-1} u_{n}\nabla\xi_{n}\cdot\nabla u_{n}+\xi_{n}^{r}u_{n}^{2})-\int _{\Omega}\xi_{n}^{r}|u_{n}|^{p}. \end{align*} Note that $|\nabla\xi_{n}(z)|\leq\frac{c}{n}$ and $\{ u_{n}\} $ is bounded in $X(\Omega)$, so \[ \int_{\Omega}\xi_{n}^{r-1}u_{n}\nabla\xi_{n}\cdot\nabla u_{n}= o(1). \] We conclude that \[ \int_{\Omega}\xi_{n}^{r}(|\nabla u_{n}|^{2}+u_{n}^{2})=\int _{\Omega}\xi_{n}^{r}|u_{n}|^{p}+ o(1)=\frac{2p} {p-2}\beta+ o(1). \] Therefore, the results follow.\newline$(iii)$ By the H\"{o}lder and Sobolev inequalities, we have \[ |{\int_{\Omega}} (\xi_{n}^{r}-1) u_{n}\varphi| \leq\Big(\int_{\Omega_{n}}|u_{n}|^{2}\Big) ^{1/2} \Big(\int_{\Omega}|\varphi|^{2}\Big) ^{1/2} \leq o(1)\| \varphi\| _{H^{1}}. \] $(iv)$ By the H\"{o}lder and Sobolev inequalities, we have \[ \Big|{\int_{\Omega}} (\xi_{n}^{r}-1) | u_{n}| ^{p-2}u_{n}\varphi\Big| \leq(\int_{\Omega_{n}}|u_{n}|^{p}) ^{\frac{p-1}{p}}( \int_{\Omega}|\varphi|^{p}) ^{1/p}\\ \leq o(1)\|\varphi\|_{H^{1}}. \] $(v)$ By the hypothesis and part $(i)$, we have \begin{align*} o(1) & =\langle J'(u_{n}),w_{n}\rangle \\ & =\langle J'(u_{n}),w_{n}\rangle -\langle J'(u_{n}),u_{n}\rangle +\langle J'(u_{n}),u_{n}\rangle \\ & =\int_{\Omega}(\xi_{n}^{r}-1) |\nabla u_{n}|^{2} +\int_{\Omega}(\xi_{n}^{r}-1) u_{n}^{2}-\int _{\Omega}(\xi_{n}^{r}-1) |u_{n}|^{p}+ o(1)\\ & =\int_{\Omega}(\xi_{n}^{r}-1) |\nabla u_{n} |^{2}+ o(1). \end{align*} Thus, \[ \big| \int_{\Omega}(\xi_{n}^{r}-1) |\nabla u_{n} |^{2}\big| =\int_{\Omega}(1-\xi_{n}^{r}) |\nabla u_{n}|^{2}= o(1). \] Therefore, by the H\"{o}lder inequality, \begin{align*} |{\int_{\Omega}} (\xi_{n}^{r}-1) \nabla u_{n}\nabla\varphi| & \leq\Big({\int_{\Omega}}(\xi_{n}^{r}-1) ^{2}| \nabla u_{n}| ^{2}\Big) ^{1/2}\| \varphi\| _{H^{1}}\\ & \leq\Big({\int_{\Omega}} (1-\xi_{n}^{r}) | \nabla u_{n}| ^{2}\Big)^{1/2} \| \varphi\| _{H^{1}}\\ & \leq o(1)\| \varphi\| _{H^{1}}. \end{align*} \end{proof} \begin{lemma}\label{p33} $(i)$ Suppose that $\{ u_{n}\} $ is a sequence in $X(\Omega)$ satisfying $u_{n}\rightharpoonup0$ weakly in $X(\Omega)$, then there is a subsequence $\{ u_{n}\} $ in $X(\Omega)$ such that $\int_{\Omega_{n}}| u_{n}| ^{p} = o(1)$ as $n\to\infty$; \newline $(ii)$ For any $\beta>0$, suppose that $\{ u_{n}\} $ is a (PS)$_{\beta}$-sequence in $X(\Omega)$ for $J$ satisfying $\int_{\Omega_{n}}| u_{n}| ^{p}= o(1)$ as $n\to\infty$, then $\{ \xi_{n} u_{n}\} $ is also a (PS)$_{\beta}$-sequence in $X(\Omega)$ for $J$. \end{lemma} \begin{proof} $(i)$ Since $u_{n}\rightharpoonup0$ weakly in $X(\Omega)$, there is a subsequence $\{ u_{n}\} $ such that $u_{n}\to u$ strongly in $L_{\rm loc}^{p}(\Omega) $, or there is a subsequence $\{ u_{n}\} $ such that \[ \int_{\Omega_{n}}| u_{n}| ^{p}= o(1), \] where $\Omega_{n}=\Omega\cap B^{N}(0;n) $. \newline $(ii)$ Let $\{ u_{n}\} $ be a (PS)$_{\beta}$-sequence in $X(\Omega)$ for $J$ satisfying $\int_{\Omega_{n}}| u_{n}| ^{p}= o(1)$ as $n\to\infty$. By Lemma \ref{p32}, we have \begin{align*} J(\xi_{n}u_{n}) & =\frac{1}{2} {\int_{\Omega}} \left[ |\nabla(\xi_{n}u_{n}) |^{2}+(\xi_{n}u_{n}) ^{2}\right] -\frac{1}{p} {\int_{\Omega}}| \xi_{n}u_{n}| ^{p}\\ & =\frac{1}{2} {\int_{\Omega}} \left[ |\nabla\xi_{n}|^{2}u_{n}^{2}+\xi_{n}^{2}(|\nabla u_{n}|^{2}+u_{n} ^{2})+2\xi_{n}u_{n}\nabla\xi_{n}\nabla u_{n}\right] -\frac{1}{p} {\int_{\Omega}} \xi_{n}^{p}| u_{n}| ^{p}\\ & =\frac{1}{2}a(u_{n}) -\frac{1}{p}b(u_{n}) + o(1) =\beta+ o(1) . \end{align*} Then for $\varphi\in X(\Omega) $, we have \begin{align*} &| \langle J'(\xi_{n}u_{n}),\varphi\rangle| \\ & =\big|\langle J'(\xi_{n}u_{n}),\varphi\rangle-\langle J' (u_{n}),\varphi\rangle+\langle J'(u_{n}),\varphi\rangle\big| \\ & =\big| \int_{\Omega}(\xi_{n}\nabla u_{n}\nabla\varphi+u_{n} \nabla\xi_{n}\nabla\varphi+\xi_{n}u_{n}\varphi-\xi_{n}^{p-1}|u_{n}|^{p-2} u_{n}\varphi) \\ &\quad -\langle J'(u_{n}),\varphi\rangle+\langle J' (u_{n}),\varphi\rangle\big| \\ & =\big| \int_{\Omega}\left[ (\xi_{n}-1)\nabla u_{n}\nabla \varphi+(\xi_{n}-1)u_{n}\varphi-(\xi_{n}^{p-1}-1)|u_{n}|^{p-2}u_{n} \varphi\right] +\langle J'(u_{n}),\varphi\rangle\big| \\ & \leq o(1)\| \varphi\| _{H^{1}} \end{align*} Thus, $J'(\xi_{n}u_{n})= o(1)$. \end{proof} Moreover, we have the following lemma. \begin{lemma} \label{p34} Let $\{ u_{n}\} $ be a (PS)-sequence in $H_{0} ^{1}(\Omega) $ for $J$ satisfying $u_{n}\rightharpoonup0$ weakly in $X(\Omega) $ and let $v_{n}=\xi_{n}u_{n}$. Then $\| u_{n}-v_{n}\| _{H^{1}}= o(1)$ as $n\to\infty$. \end{lemma} \begin{proof} Note that \begin{align*} a(u_{n}-v_{n}) & =\langle u_{n}-v_{n},u_{n}-v_{n} \rangle _{H^{1}}\\ & =a(u_{n}) +a(v_{n}) -2\langle u_{n} ,v_{n}\rangle _{H^{1}}\\ & =2a(u_{n}) -2\langle u_{n},v_{n}\rangle _{H^{1} }+ o(1) . \end{align*} Thus, it suffices to show that \[ a(u_{n}) =\langle u_{n},v_{n}\rangle _{H^{1}}+ o(1) . \] We have \begin{align*} \langle u_{n},v_{n}\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)-sequence in $H_{0}^{1}(\Omega) $ for $J$, so \[ \int_{\Omega}u_{n}\nabla u_{n}\nabla\xi_{n}= o(1) . \] Hence, \[ \langle u_{n},v_{n}\rangle _{H^{1}}=\int_{\Omega}\xi_{n}\left[ | \nabla u_{n}| ^{2}+(u_{n}) ^{2}\right] +o( 1) . \] By Lemma \ref{p32} $(i)$, $(ii) $ and Lemma \ref{p33} $(i)$, we have \[ \langle u_{n},v_{n}\rangle _{H^{1}} =\int_{\Omega}\xi_{n}\left[ | \nabla u_{n}| ^{2}+(u_{n}) ^{2}\right] +o(1) =a(u_{n}) + o(1) . \] \end{proof} \noindent\textbf{Bibliographical notes:} The (PS)-sequences were originally introduced by Palais-Smale \cite{PS}. Lemma \ref{p3} is from Br\'{e}zis \cite[p. 35]{B}. Lemma \ref{p4} is from Zeidler \cite[II/A, p. 303]{Z}. Lemma \ref{p9} is from Bahri-Lions \cite{BaLi}. Lemma \ref{p12} is from Grisvard \cite[p. 24]{G}. \section{Palais-Smale Decomposition Theorems} In this section, we present the Palais-Smale decomposition theorem in $H_{0}^{1}(\Omega)$ for $J$. This is the concentration-compactness method of P. L. Lions. \begin{theorem}[Palais-Smale Decomposition Theorem in $\mathbb{R}^{N})$] \label{d1} Let $\Omega$ be strictly large domain (see Definition \ref{f4}) in $\mathbb{R}^{N}$ and let $\{ u_{n}\} $ be a (PS)$_{\beta}$-sequence in $H_{0}^{1}(\Omega)$ for $J$. Then there are a subsequence $\{ u_{n}\} $, a positive integer $m$, sequences $\{z_{n}^{i}\}_{n=1}^{\infty}$ in $\mathbb{R}^{N}$, a function $\bar{u}\in H_{0}^{1}(\Omega)$, and $0\neq w^{i}\in H^{1}(\mathbb{R}^{N}) $ for $1\leq i\leq m$ such that \begin{gather*} |z_{n}^{i}|\to\infty,\quad \text{for }i=1,2,\dots,m,\\ -\Delta\bar{u}+\bar{u}=|\bar{u}|^{p-2}\bar{u}\quad\mbox{in }\Omega,\\ -\Delta w^{i}+w^{i}=\mid w^{i}|^{p-2}w^{i}\quad\mbox{in }\mathbb{R}^{N}, \end{gather*} and \begin{gather*} u_{n}=\bar{u}+\sum_{i=1}^m w^{i}(\cdot-z_{n} ^{i}) + o(1)\;\text{strongly}\quad\text{}\text{in}\quad\text{}H^{1}( \mathbb{R}^{N}) ,\\ a(u_{n})=a(\bar{u})+\sum_{i=1}^m a(w^{i})+ o(1),\\ b(u_{n})=b(\bar{u})+\sum_{i=1}^m b(w^{i})+ o(1),\\ J(u_{n})=J(\bar{u})+\sum_{i=1}^m J(w^{i})+ o(1). \end{gather*} In addition, if $u_{n}\geq0$, then $\bar{u}\geq0$ and $w^{i}\geq0$ for each $1\leq i\leq m$. \end{theorem} \begin{proof} \textbf{Step 0}. Since $\{ u_{n}\} $ is a (PS)$_{\beta}$-sequence in $H_{0}^{1}(\Omega) $ for $J$, by Lemma \ref{p30} there is a $c>0$ such that $\| u_{n}\| _{H^{1}}\leq c$. In the following proof of this theorem, we fix such a $c$. There is a subsequence $\{ u_{n}\} $ and a $\bar{u}$ in $H_{0}^{1}(\Omega)$ such that $u_{n}\rightharpoonup\bar{u}$ weakly in $H_{0}^{1}(\Omega)$ and $\bar{u}$ solves \[ -\Delta\bar{u}+\bar{u}=|\bar{u}|^{p-2}\bar{u}\quad\mbox{in }\Omega. \] Suppose that $u_{n}\to\bar{u}$ strongly in $H_{0}^{1}(\Omega)$, then we have $u_{n}=\overline{u}+ o(1) $ strongly in $H_{0}^{1}( \Omega) $, $a(u_{n}) =a(\overline{u}) + o(1) $, $b(u_{n}) =b(\overline{u}) + o(1) $, $J(u_{n}) =J(\overline{u}) + o(1) $.\newline \textbf{Step 1}. Suppose that $u_{n} \nrightarrow\bar{u}$ strongly in $H_{0}^{1}(\Omega)$. Let \[ u_{n}^{1}=u_{n}-\bar{u}\quad \text{for }n=1,2,\dots. \] By Lemma \ref{p8}, $\{u_{n}^{1}\}$ is a (PS)$_{(\beta-J(\bar{u}))}$-sequence in $H_{0}^{1}(\Omega)$ for $J$. \begin{itemize} \item[(1-0)] $ {\int_{B^{N}(0;1) }} |w_{n}^{1}(z)|^{2}dz\geq\frac{d_{1}}{2}$ for some constant $d_{1}>0$ and $n=1,2,\dots$, where $w_{n}^{1}(z)=u_{n}^{1}(z+y_{n}^{1}) $ for some $\{ y_{n}^{1}\} \subset\mathbb{R}^{N}$: since $\{ u_{n}^{1}\} $ is bounded, $J'(u_{n}^{1})= o(1)$, and $u_{n}^{1}\nrightarrow0$ strongly in $H_{0}^{1}(\Omega)$. By Lemma \ref{p9} there is a subsequence $\{ u_{n}^{1}\} $, a constant $d_{1}>0$ such that \[ Q_{n}^{1}=\sup_{z\in\mathbb{R}^{N}}\int_{z+B^{N}(0;1)}|u_{n}^{1}|^{2}\geq d_{1}\quad\text{for } n=1,2,\dots. \] Take $\{ y_{n}^{1}\} $ in $\mathbb{R}^{N}$ such that \[ \int_{y_{n}^{1}+B^{N}(0;1)}|u_{n}^{1}(z)|^{2}dz\geq\frac{d_{1}}{2}. \] Let $w_{n}^{1}(z)=u_{n}^{1}(z+y_{n}^{1}) $, then \[ \int_{B^{N}(0;1)}|w_{n}^{1}(z)|^{2}dz\geq\frac{d_{1}}{2}\quad\text{for } n=1,2,\dots. \] \item[(1-1)] $u_{n}(z)=\bar{u}(z)+w_{n}^{1}(z-y_{n}^{1}) $ in $H^{1}(\mathbb{R}^{N})$. \item[(1-2)] $\Vert w_{n}^{1}\Vert_{H^{1}(\mathbb{R}^{N})}\leq c$ for $n=1,2,\dots$ and $\| w^{1}\| _{H^{1}}\leq c$, where $w_{n}^{1}\rightharpoonup w^{1}$ weakly in $H^{1}(\mathbb{R}^{N})$: by Lemma \ref{p4} $(iii) $, \[ \| w_{n}^{1}\| _{H^{1}}^{2}=\| u_{n}^{1}\| _{H^{1}}^{2}=\| u_{n}\| _{H^{1}} ^{2}-\| \bar{u}\| _{H^{1}}^{2}+ o(1)\leq c^{2}+ o(1), \] we have $\Vert w_{n}^{1}\Vert_{H^{1}(\mathbb{R}^{N})}\leq c$ for $n=1,2,\dots$. Then there is a subsequence $\{w_{n}^{1}\}$ and a $w^{1}$ in $H^{1}(\mathbb{R}^{N})$ such that $w_{n}^{1}\rightharpoonup w^{1}$ weakly in $H^{1}(\mathbb{R}^{N})$. By Lemma \ref{p4} $(i)$, we have \[ \| w^{1}\| _{H^{1}}\leq\liminf_{n\to\infty}\| w_{n} ^{1}\| _{H^{1}}\leq c. \] \item[(1-3)] $\{w_{n}^{1}\}$ is a (PS)$_{(\beta-J(\bar{u})) }$-sequence in $H^{1}(\mathbb{R}^{N})$ for $J$: note that $J' (u_{n}^{1})= o(1) $ in $H^{-1}(\Omega) $. Because $\Omega$ is a strictly large domain, $(1$-7) below and Theorem \ref{p271}, we have for every $\varphi\in H_{0}^{1}(\mathbb{R}^{N})$, \[ \langle J'(w_{n}^{1}) ,\varphi\rangle =\int_{\mathbb{R}^{N}}\nabla w_{n}^{1}\nabla\varphi+w_{n}^{1}\varphi -\int_{\mathbb{R}^{N}}| w_{n}^{1}| ^{p-2}w_{n}^{1}\varphi = o(1) \] Therefore, $J'(w_{n}^{1}) = o(1)$ strongly in $H^{-1}(\mathbb{R}^{N}) $. Moreover, we have \[ J(w_{n}^{1}) =J(u_{n}^{1}(z+y_{n}^{1}) )=J( u_{n}^{1}) =(\beta-J(\bar{u})) + o(1). \] \item[(1-4)] $-\Delta w^{1}+w^{1}-|w^{1}|^{p-2}w^{1}=0$ in $\mathbb{R}^{N}$ : by Theorem \ref{c2} $(i)$ below. \item[(1-5)] $w^{1}\not \equiv 0$: by the Rellich-Kondrakov theorem \ref{p24} and $(1-0)$, we have \[ \int_{B^{N}(0;1) }|w^{1}|^{2}=\lim_{n\to\infty} \int_{B^{N}(0;1) }|w_{n}^{1}|^{2}\geq\frac{d_{1}}{2}, \] thus $w^{1}\not \equiv 0$. \item[(1-6)] By $(1-2)$, $(1-4)$, $(1-5)$, and Lemma \ref{p11}, $\delta>0 $ exists such that \[ \Vert w^{1}\Vert_{H^{1}(\mathbb{R}^{N})}\geq\Vert w^{1}\Vert_{L^{2} (\mathbb{R}^{N})}>\delta. \] Therefore, \[ J(w^{1})=(\frac{1}{2}-\frac{1}{p}) a(w^{1})>(\frac{1} {2}-\frac{1}{p}) \delta^{2}=\delta'. \] \item[(1-7)] $|y_{n}^{1}|\to\infty$: otherwise, $R>0$ exists such that $y_{n}^{1}+B^{N}(0;1) \subset B^{N}(0;R) \; $for $n=1,2,\dots$. Then by $(1-0)$, we have \[ 0=\lim_{n\to\infty}\int_{B^{N}(0;R) }|u_{n}^{1}|^{2} \geq\overline{\lim_{n\to\infty}}\int_{y_{n}^{1}+B^{N}( 0;1) }|u_{n}^{1}|^{2}\geq\frac{d_{1}}{2}, \] which is a contradiction. \item[(1-8)] $a(u_{n})=a(\bar{u})+a(w_{n}^{1})+ o(1):$ since $u_{n} \rightharpoonup\bar{u}$ weakly in $H^{1}(\mathbb{R}^{N})$, by Lemma \ref{p4} $(iii)$, we have \[ a(u_{n})-a(\bar{u}) =a(u_{n}-\bar{u})+ o(1) =a(u_{n}^{1}) + o(1) =a(w_{n}^{1})+ o(1) . \] Thus, $a(u_{n})=a(\bar{u})+a(w_{n}^{1})+ o(1)$. \item[(1-9)] $b(u_{n})=b(\bar{u})+b(w_{n}^{1})+ o(1)$: since $u_{n} \to\bar{u}$ a.e. in $\Omega$ and $\{ u_{n}\} $ is bounded in $L^{p}(\Omega) $, by Lemma \ref{p7} $(i) $, we have \[ b(u_{n})-b(\bar{u}) =b(u_{n}-\bar{u})+ o(1) =b(u_{n}^{1}) + o(1) =b(w_{n}^{1})+ o(1) . \] Thus, $b(u_{n})=b(\bar{u})+b(w_{n}^{1})+ o(1)$. \item[(1-10)] $J(u_{n})=J(\bar{u})+J(w_{n}^{1})+ o(1)$: by $(1-8)$ and $(1-9)$, we have \[ J(u_{n})=J(\bar{u})+J(w_{n}^{1})+ o(1). \] \end{itemize} \textbf{Step 2}. Suppose that $w_{n}^{1}(z)\nrightarrow w^{1}(z)$ strongly in $H^{1}(\mathbb{R}^{N})$. Let \[ u_{n}^{2}(z)=w_{n}^{1}(z)-w^{1}(z). \] We have $u_{n}^{2}\rightharpoonup0$ weakly in $H^{1}(\mathbb{R}^{N})$ but $u_{n}^{2}\nrightarrow0$ strongly in $H^{1}(\mathbb{R}^{N})$. \begin{itemize} \item[(2-0)] $ {\int_{B^{N}(0;1) }} |w_{n}^{2}(z)|^{2}dz\geq\frac{d_{2}}{2}$ for some constant $d_{2}>0$ and $n=1,2,\dots$, where $w_{n}^{2}(z)=u_{n}^{2}(z+y_{n}^{2}) $ for some $\{ y_{n}^{2}\} \subset\mathbb{R}^{N}$: since $\{ u_{n}^{2}\} $ is bounded, $J'(u_{n}^{2})= o(1)$, and $u_{n}^{2}\nrightarrow0$ strongly in $H^{1}(\mathbb{R}^{N})$, by Lemma \ref{p9} there are a subsequence $\{ u_{n}^{2}\} $, and a constant $d_{2}>0$ such that \[ Q_{n}^{2}=\sup_{z\in\mathbb{R}^{N}}\int_{z+B^{N}(0;1) } |u_{n}^{2}(z) |^{2}dz\geq d_{2}\quad\mbox{for } n=1,2,\dots. \] \end{itemize} For $n=1,2,\dots$, take $\{ y_{n}^{2}\} $ in $\mathbb{R}^{N}$ such that \[ \int_{y_{n}^{2}+B^{N}(0;1) }|u_{n}^{2}(z)|^{2}dz\geq\frac{d_{2} }{2}\quad\text{for } n=1,2,\dots. \] Let $w_{n}^{2}(z)=u_{n}^{2}(z+y_{n}^{2})$, then \[ \int_{B^{N}(0;1) }|w_{n}^{2}(z)|^{2}dz\geq\frac{d_{2}} {2}\quad \text{for } n=1,2,\dots. \] As in Step 1, we have the following results. \begin{itemize} \item[(2-1)] $u_{n}(z)=\bar{u}(z)+w^{1}(z-y_{n}^{1}) +w_{n}^{2}(z-y_{n}^{1}-y_{n}^{2}) $ in $H^{1}(\mathbb{R}^{N})$; \item[(2-2)] $\| w_{n}^{2}\| _{H^{1}}\leq c$ for $n=1,2,\dots$ and $\| w^{2}\| _{H^{1}}\leq c$, where $w_{n}^{2}\rightharpoonup w^{2}$ weakly in $H^{1}(\mathbb{R}^{N})$; \item[(2-3)] $\{w_{n}^{2}\}$ is a (PS)-sequence in $H^{1}(\mathbb{R}^{N})$ for $J$; \item[(2-4)] $-\Delta w^{2}+w^{2}-|w^{2}|^{p-2}w^{2}=0$ in $\mathbb{R}^{N}$; \item[(2-5)] $w^{2}\not \equiv 0$; \item[(2-6)] $\Vert w^{2}\Vert_{L^{2}(\mathbb{R}^{N})}>\delta$ and $J(w^{2})>\delta'$; \item[(2-7)] $|y_{n}^{2}|\to\infty$; \item[(2-8)] $a(u_{n})=a(\bar{u})+a(w^{1})+a(w_{n}^{2})+ o(1)$: since \[ u_{n}^{2}(z) =w_{n}^{1}(z) -w^{1}(z) \rightharpoonup 0, \] we have \[ a(w_{n}^{2}) =a(u_{n}^{2}) =a(w_{n} ^{1}) -a(w^{1}) + o(1). \] Further by $(1-8) $, we have \[ a(u_{n})-a(\bar{u})=a(w_{n}^{1})+ o(1) =a(w^{1}) +a(w_{n}^{2}) + o(1) . \] \item[(2-9)] $b(u_{n})=b(\bar{u})+b(w^{1})+b(w_{n}^{2})+ o(1)$; \item[(2-10)] $J(u_{n})=J(\bar{u})+J(w^{1})+J(w_{n}^{2})+ o(1)$. \end{itemize} Continuing this process, we arrive at the $m$-th step. \begin{itemize} \item[$(m$-0)] ${\int_{B^{N}(0;1) }} |w_{n}^{m}(z)|^{2}dz\geq\frac{d_{m}}{2}$ for some constant $d_{m}>0$ and$\;n=1,2,\dots$, where $w_{n}^{m}(z)=u_{n}^{m}(z+y_{n}^{m}) $ for some $\{ y_{n}^{m}\} \subset\mathbb{R}^{N}$; \item[$(m$-1)] $u_{n}(z)=\bar{u}(z)+\underset{i=1} {\overset{m-1}{\sum}}w^{i}(z-z_{n}^{i}) +w_{n}^{m}( z-z_{n}^{m}) $ in $H^{1}(\mathbb{R}^{N})$, where $z_{n}^{i}=y_{n} ^{1}+\dots+y_{n}^{i}$ for $i=1,2,\dots,m:$ since \[ w_{n}^{m}(z)=u_{n}^{m}(z+y_{n}^{m})=w_{n}^{m-1}(z+y_{n}^{m})-w^{m-1} (z+y_{n}^{m}), \] thus \[ w_{n}^{m}(z)+w^{m-1}(z+y_{n}^{m})=w_{n}^{m-1}(z+y_{n}^{m}). \] Continuing this way, we obtain \begin{align*} &w_{n}^{m}(z)+w^{m-1}(z+y_{n}^{m})+\dots+w^{1}(z+y_{n}^{2}+\dots+y_{n}^{m})\\ &=w_{n}^{1}(z+y_{n}^{2}+\dots+y_{n}^{m})\\ &=u_{n}^{1}(z+y_{n}^{1}+y_{n}^{2}+\dots+y_{n}^{m}) \end{align*} \item[$(m$-2)] $\| w_{n}^{m}\| _{H^{1}}\leq c$ for $n=1,2,\dots$ and $\| w^{m}\| _{H^{1}}\leq c$, where $w_{n}^{m}\rightharpoonup w^{m}$ weakly in $H^{1}(\mathbb{R}^{N})$; \item[$(m$-3)] $\{w_{n}^{m}\}$ is a (PS)-sequence in $H^{1}(\mathbb{R}^{N})$ for $J$; \item[$(m$-4)] $-\Delta w^{m}+w^{m}-|w^{m}|^{p-2}w^{m}=0$ in $\mathbb{R}^{N}$; \item[$(m$-5)] $w^{m}\not \equiv 0$; \item[$(m$-6)] $\| w^{m}\| _{L^{2}(\mathbb{R}^{N}) }>\delta$ and $J(w^{m})>\delta'$; \item[$(m$-7)] $|y_{n}^{i}|=$ $|z_{n}^{i}-z_{n}^{i-1}|\to\infty$ and $|z_{n}^{i}|\to\infty$, for each $i=1,2,\dots,m:$ we show it by induction on $i$. For $i=1$, $| z_{n}^{1}| =| y_{n} ^{1}| \to\infty$. Assume that $|z_{n}^{i}|\to\infty$, for $i=1,2,\dots,k$, for some $k\delta$, $R>0$ exists such that \[ z_{n}^{k+1}+B^{N}(0;R) \subset B^{N}(0;2R) \] and \[ {\int_{B^{N}(0;R) }}|w^{k+1}(z) |^{2}\geq(\frac{\delta}{2}) ^{2}. \] We have \begin{align*} (\frac{\delta}{2}) ^{2}&\leq \int_{B^{N}(0;R) }{\int_{B^{N}(0;R) }}|w^{k+1}(z) |^{2}\\ &=\lim_{n\to\infty}{\int_{B^{N}(0;R) }} |u_{n}^{1}(z+z_{n}^{k+1})|^{2}dz\\ &\leq\lim_{n\to\infty} {\int_{B^{N}(0;2R) }} |u_{n}^{1}(z)|^{2}=0, \end{align*} which is a contradiction. By the induction hypothesis, we have \[ |z_{n}^{i}|\to\infty\quad \text{for }i=1,2,\dots,m. \] \item[$(m$-8)] $a(u_{n})=a(\bar{u})+\sum_{i=1}^{m-1} a(w^{i})+a(w_{n}^{m})+ o(1)$; \item[$(m$-9)] $b(u_{n})=b(\bar{u})+\sum_{i=1}^{m-1} b(w^{i})+b(w_{n}^{m})+ o(1)$; \item[$(m$-10)] $J(u_{n})=J(\bar{u})+\sum_{i=1}^{m-1} J(w^{i})+J(w_{n}^{m})+ o(1)$. \end{itemize} By the Archimedean principle, $l\in\mathbb{N}$ exists such that $l\delta ^{2}>\beta$. Then after step $(l+1)$, we obtain \[ a(u_{n})=a(\bar{u})+a(w^{1})+a(w^{2})+\dots+a(w^{l})+a(w_{n}^{l+1})+ o(1). \] Since $a(w_{n}^{l+1})\geq0$, $a(\bar{u})>0$, and $a(w^{i})>\delta^{2}$ for $i=1,2,\dots,l$, we have $\beta+ o(1)\geq l\delta^{2}>\beta$, which is a contradiction. Therefore, there is an $m\in\mathbb{N}$, such that $w_{n} ^{m}(z)=w^{m}(z)+ o(1)$ strongly in $H^{1}(\mathbb{R}^{N})$, $w_{n} ^{i}(z)=w^{i}(z)+ o(1)$ weakly, and $w_{n}^{i}(z)\neq w^{i}(z)+ o(1)$ strongly in $H^{1}(\mathbb{R}^{N})$ for $i=1,2,\dots m-1$. Then we have \begin{itemize} \item[$(sm$-0)] ${\int_{B^{N}(0;1) }} |w_{n}^{m}(z)|^{2}dz\geq\frac{d_{m}}{2}$ for some constant $d_{m}>0$ and $n=1,2,\dots$, where $w_{n}^{m}(z)=u_{n}^{m}(z+y_{n}^{m})$ for some $\{ y_{n}^{m}\} \subset\mathbb{R}^{N}$; \item[$(sm$-1)] $u_{n}(z)=\bar{u}(z)+\underset{i=1}{\overset {m}{\sum}}w^{i}(z-z_{n}^{i}) + o(1) $ strongly in $H^{1}(\mathbb{R}^{N})$, where $z_{n}^{i}=y_{n}^{1}+\dots+y_{n}^{i}$ for $i=1,2,\dots,m$; \item[$(sm$-2)] $\| w_{n}^{m}\| _{H^{1}}\leq c$ $\text{for\;} n=1,2,\dots$ and $\| w^{m}\| _{H^{1}}\leq c$, where $w_{n}^{m}\rightharpoonup w^{m}$ weakly in $H^{1}(\mathbb{R}^{N})$; \item[$(sm$-3)] $\{w_{n}^{m}\}$ is a (PS)-sequence in $H^{1}(\mathbb{R} ^{N})$ for $J$; \item[$(sm$-4)] $-\Delta w^{m}+w^{m}-|w^{m}|^{p-2}w^{m}=0$ in $\mathbb{R}^{N}$; \item[$(sm$-5)] $w^{m}\not \equiv 0$; \item[$(sm$-6)] $\| w^{m}\| _{L^{2}(\mathbb{R}^{N})}>\delta$ and $J(w^{m})>\delta'$; \item[$(sm$-7)] $|y_{n}^{i}|=$ $|z_{n}^{i}-z_{n}^{i-1}|\to\infty$ and $|z_{n}^{i}|\to\infty$, for each $i=1,2,\dots,m$; \item[$(sm$-8)] $a(u_{n})=a(\bar{u})+\sum_{i=1}^m a(w^{i})+ o(1)$; \item[$(sm$-9)] $b(u_{n})=b(\bar{u})+\sum_{i=1}^m b(w^{i})+ o(1)$; \item[$(sm$-10)] $J(u_{n})=J(\bar{u})+\underset{i=1} {\overset{m}{\sum}}J(w^{i})+ o(1)$. \end{itemize} Finally, suppose $u_{n}\geq0$ for $n=1,2,\dots$. Then\newline $(i)$ Since $u_{n}\rightharpoonup\bar{u}$ weakly in $H_{0}^{1}(\Omega)$. By Lemma \ref{p4} $(ii)$, there is a subsequence $\{u_{n}\}$ such that $\,u_{n}\to\bar{u}$ a.e. in $\Omega$. Thus, $\bar{u}\geq0$. \newline $(ii)$ Since $w_{n}^{1}(z)=u_{n}(z+y_{n}^{1})-\bar{u}(z+y_{n} ^{1})\rightharpoonup w^{1}(z)$ weakly in $H^{1}(\mathbb{R}^{N})$ and $\bar {u}(z+y_{n}^{1})\rightharpoonup0$ weakly in $H^{1}(\mathbb{R}^{N})$. Thus, $u_{n}(z+y_{n}^{1})\to w^{1}(z)$ a.e. in $\Omega$, or $w^{1}\geq 0$. \newline $(iii)$ Continuing this process, we obtain $w^{i}\geq0$ for each $i=1,2,\dots,m$. \end{proof} We have the following useful corollary. \begin{corollary} \label{d10} Let $\Omega$ be a strictly large domain in $\mathbb{R}^{N}$. If $\{ u_{n}\} $ is a positive ($PS$)$_{\beta}$-sequence in $H_{0}^{1}(\Omega) $ for $J$.\newline $(i)$ If $\beta\neq j\alpha(\mathbb{R}^{N}) $ for each $j\in\mathbb{N}$, then there is a positive solution $\overline{u}$ of \eqref{E1} in $\Omega $; \newline$(ii)$ If $\alpha(\mathbb{R}^{N}) <\beta <2\alpha(\mathbb{R}^{N}) $, then $\{ u_{n}\} $ contains a strongly convergent subsequence. \end{corollary} \begin{proof} By Theorem \ref{d1}, we have \[ J(u_{n}) =J(\overline{u}) +\sum{i=1} ^{m}J(w^{i})+ o(1). \] By Corollary \ref{d4} below, the positive solutions of \eqref{E1} in $\mathbb{R}^{N}$ are unique, and we obtain $J(w^{i})=\alpha( \mathbb{R}^{N}) $ for each $i$. Thus, we have \[ \beta=J(\overline{u}) +mJ(w^{i})+ o(1). \] $(i)$ If $\beta\neq j\alpha(\mathbb{R}^{N}) $ for each $j\in\mathbb{N}$, then $J(\overline{u})\neq0$, or $\overline{u}\neq0$. By Theorem \ref{c2} $(i)$ below, there is a positive solution $\overline{u}$ of \eqref{E1} in $\Omega$.\newline $(ii)$ Recall that we always have $\beta\geq\alpha(\Omega) \geq\alpha(\mathbb{R} ^{N}) $. Suppose that $m\geq1$ and $\alpha(\mathbb{R} ^{N}) <\beta<2\alpha(\mathbb{R}^{N}) $, then $J(\overline{u})\neq0$ or $J(\overline{u})\geq\alpha(\Omega) $. Thus, \[ 2\alpha(\mathbb{R}^{N}) >\beta+ o(1)=J(\overline {u}) +m\alpha(\mathbb{R}^{N}) \geq(m+1)\alpha( \mathbb{R}^{N}) . \] This is a contradiction. Hence, $m=0$. By the proof of Theorem \ref{d1}, we have \[ u_{n}=\overline{u}+ o(1)\quad\text{strongly in }H_{0}^{1}(\Omega). \] \end{proof} \begin{remark}\label{d11} \rm Note that if we replace a strictly large domain by a domain in Theorem \ref{d1}, then the theorem may fail. Let $\mathbf{A}_{0}^{r}$ be an upper semi-strip with sufficiently large $r$, then $\alpha(\mathbb{R} ^{N})<\alpha(\mathbf{A}_{0}^{r})<2\alpha(\mathbb{R}^{N})$. By the Esteban-Lions theorem \ref{n3}, \eqref{E1} in $\mathbf{A}_{0}^{r}$ admits only trivial solution, but if Theorem \ref{d1} holds, by Corollary \ref{d10}, \eqref{E1} in $\mathbf{A}_{0}^{r}$ admits a positive solution, a contradiction. \end{remark} \begin{definition} \label{d2} \rm A domain $\Theta$ in $\mathbb{R}^{N}$ is a periodic domain if a partition $\{Q_{m}\}_{m=0}^{\infty}$ of $\Theta$ and points $\{z_{m} \}_{m=1}^{\infty}$ in $\mathbb{R}^{N}$ exist, satisfying the following conditions:\newline $(i)$ $\{z_{m}\}_{m=1}^{\infty}$ forms a subgroup of $\mathbb{R}^{N}$; \newline $(ii)$ $Q_{0}$ is bounded;\newline $(iii)$ $Q_{m}=z_{m}+Q_{0}$ for each $m$. \end{definition} Typical examples of periodic domains are the infinite strip $\mathbf{A}^{r}$, the infinite hollow strip $\mathbf{A}^{r_{1},r_{2}}$, and the whole space $\mathbb{R}^{N}$. Similarly, we have the Palais-Smale decomposition theorem in $H_{0}^{1}(\Omega)$ for $J$ in a periodic domain in $\Theta\subset \mathbb{R}^{N}$. \begin{theorem}[Palais-Smale Decomposition Theorem in a Periodic Domain] \label{d3} Let $\Omega$ be a strictly large domain in $\Theta$ and let $\{u_{n}\} $ be a positive (PS)$_{\beta}$-sequence in $H_{0}^{1}(\Omega) $ for $J$. Then there are a subsequence $\{ u_{n}\} $, a positive integer $m$, a subsequence $\{ z_{n}^{i}\} _{n=1}^{\infty}$ of $\{ z_{m}\} _{m=1}^{\infty}$ in $\Theta$, and a function $\bar {u}\in H_{0}^{1}(\Omega)$, and $0\neq w^{i}\in H^{1}(\Theta) $, for $1\leq i\leq m$ such that \newline $(i)$ $|z_{n}^{i}|\to,\infty\ \text{for}$ $i=1,2,\dots,m$; \newline $(ii)$ $-\Delta\bar{u}+\bar {u}=\mid\bar{u}\mid^{p-2}\bar{u}$ in $\Omega$; \newline $(iii)$ $-\Delta w^{i}+w^{i}=| w^{i}|^{p-2}w^{i}$ in $\Theta$; \newline $(iv)$ $u_{n}=\bar {u}+\sum_{i=1}^m w^{i}(\cdot-z_{n}^{i}) + o(1)\;\text{strongly}$ in $H^{1}(\Theta)$; \newline $(v)$ $a(u_{n})=a(\bar{u})+\sum_{i=1}^m a(w^{i})+ o(1)$; \newline $(vi)$ $b(u_{n})=b(\bar{u})+\sum_{i=1}^{m} b(w^{i})+ o(1)$; \newline $(vii)$ $J(u_{n})=J(\bar{u})+\sum_{i=1}^{m} J(w^{i})+ o(1)$.\newline In addition, if $u_{n}\geq0$, then $\bar{u}\geq0$ and $w^{i}\geq0$ for each $1\leq i\leq m$. \end{theorem} \begin{proof} The proof is similar to those of Theorem \ref{d1}: see Lien-Tzeng-Wang \cite{LTW}. Note that instead of \[ Q_{n}=\sup_{z\in\mathbb{R}^{N}}\int_{z+B^{N}(0;1)}|u_{n}(z)|^{2}dz \] we use \[ Q_{n}^{r}=\sup_{y\in\mathbb{R}}\int_{(0,y)+\mathbf{A}_{-1,1}^{r}} |u_{n}(z)|^{2}dz, \] where $\mathbf{A}_{-1,1}^{r}=\{(x,y)\in\mathbf{A}^{r}\ |\ -10$. Moreover, $\alpha_{\gamma}(\Omega)$ is a (PS)-value in $X(\Omega)$ for $J$. \begin{theorem} \label{i1} $\alpha_{\gamma}(\Omega)$ is a (PS)-value in $X(\Omega)$ for $J$. \end{theorem} \begin{proof} Let $\{u_{n}\}$ in $X(\Omega)$ be a maximizing sequence of $\gamma(\Omega)$. Then $a(u_{n})=1$ for $n=1,2,\dots$, and \[ {\int_{\Omega}}|u_{n}|^{p}=\gamma(\Omega)^{p}+ o(1)\quad \text{as }n\to\infty. \] Let $v_{n}=\gamma(\Omega)^{\frac{p}{2-p}}u_{n}$ for each $n=1,2,\dots$. Then we have \begin{gather*} a(v_{n})={\int_{\Omega}} (|\nabla v_{n}|^{2}+v_{n}^{2})=\gamma(\Omega)^{\frac{2p}{2-p}}\quad \text{for each }n=1,2,\dots,\\ b(v_{n})= {\int_{\Omega}} |v_{n}|^{p}=\gamma(\Omega)^{\frac{2p}{2-p}}+ o(1)\quad \text{as } n\to \infty, \end{gather*} and \begin{align*} J(v_{n}) & =\frac{1}{2}a(v_{n})-\frac{1}{p}b(v_{n})\\ & =(\frac{1}{2}-\frac{1}{p})\gamma(\Omega)^{\frac{2p}{2-p}}+ o(1)\quad\text{as } n\to\infty\\ & =\alpha_{\gamma}(\Omega)+ o(1)\quad \text{as }n\to\infty. \end{align*} For each $n=1,2,\dots$ and $\varphi\in X(\Omega)$, denote \[ l_{n}(\varphi)=\int_{\Omega}|v_{n}|^{p-2}v_{n}\varphi. \] Let $\phi\in X(\Omega)$ satisfy $\Vert\phi\Vert_{H^{1}}=1$. Then $\gamma(\Omega)\geq\Vert\phi\Vert_{L^{p}}$ and \begin{align*} |l_{n}(\phi)| & =\big| {\int_{\Omega}}| v_{n}|^{p-2}v_{n}\phi\big|\leq\Big( {\int_{\Omega}}|v_{n}|^{p}\Big) ^{(p-1)/p} \Big({\int_{\Omega}}|\phi|^{p}\Big)^{1/p}\\ & \leq\gamma(\Omega)^{\frac{2p-2}{2-p}}\gamma(\Omega)+ o(1)=\gamma (\Omega)^{\frac{p}{2-p}}+ o(1)\quad\text{as }n\to\infty. \end{align*} Thus, \[ \Vert l_{n}\Vert_{X^{-1}}\leq\gamma(\Omega)^{\frac{p}{2-p}}+ o(1)\quad \text{as }n\to\infty. \] Furthermore, \[ l_{n}\big(\frac{v_{n}}{\Vert v_{n}\Vert_{H^{1}}}\big) =\frac{\int_{\Omega}|v_{n}|^{p}}{\Vert v_{n}\Vert_{H^{1}}}=\frac{\gamma (\Omega)^{2p/(2-p)}}{\gamma(\Omega)^{p/(2-p)}}+ o(1)=\gamma(\Omega)^{\frac {p}{2-p}}+ o(1) \] as $n\to\infty$. We conclude that \[ \Vert l_{n}\Vert_{X^{-1}}=\gamma(\Omega)^{\frac{p}{2-p}} + o(1)\quad\mbox{as }n\to\infty. \] Since $l_{n}$ is a continuous linear functional in $X(\Omega)$, by the Riesz representation theorem, for each $n$, $w_{n}\in X(\Omega)$ exists such that \[ l_{n}(\varphi)=\langle w_{n},\varphi\rangle _{H^{1}} =\int_{\Omega}(\nabla w_{n}\cdot\nabla\varphi+w_{n}\varphi)\quad\text{for each } \varphi\in X(\Omega), \] and $\Vert w_{n}\Vert_{H^{1}}=\Vert l_{n}\Vert_{X^{-1}}$. Since \[ \langle w_{n},v_{n}\rangle_{H^{1}}=l_{n}(v_{n}) =\int_{\Omega }| v_{n}| ^{p}=\gamma(\Omega)^{\frac{2p}{2-p}}+ o(1)\quad \text{as } n\to\infty, \] we obtain \begin{align*} \Vert v_{n}-w_{n}\Vert_{H^{1}}^{2} & =\langle v_{n},v_{n}\rangle_{H^{1}}-2\langle v_{n},w_{n}\rangle_{H^{1}} +\langle w_{n},w_{n}\rangle_{H^{1}}\\ & =\Vert v_{n}\Vert_{H^{1}}^{2}-2\langle v_{n},w_{n}\rangle_{H^{1}}+\Vert w_{n}\Vert_{H^{1}}^{2}\\ & =\gamma(\Omega)^{\frac{2p}{2-p}}-2\gamma(\Omega)^{\frac{2p}{2-p}} +\gamma(\Omega)^{\frac{2p}{2-p}}+ o(1)\\ & = o(1)\quad \text{as } n\to\infty. \end{align*} For $\varphi\in X(\Omega)$ satisfying $\Vert\varphi\Vert_{H^{1}}=1$, we have \begin{align*} \langle J'(v_{n}),\varphi\rangle & = {\int_{\Omega}} (\nabla v_{n}\cdot\nabla\varphi+v_{n}\varphi)- {\int_{\Omega}} |v_{n}|^{p-2}v_{n}\varphi\\ & =\langle v_{n},\varphi\rangle_{H^{1}}-\langle w_{n},\varphi\rangle_{H^{1} }=\langle v_{n}-w_{n},\varphi\rangle_{H^{1}}, \end{align*} so \[ |\langle J'(v_{n}),\varphi\rangle|\leq\Vert v_{n}-w_{n}\Vert_{H^{1}}. \] We conclude that \[ J'(v_{n})= o(1)\quad\text{strongly in }X^{-1}(\Omega)\quad\text{as }n\to\infty. \] \end{proof} \noindent (B) Consider the Nehari minimizing problem \[ \alpha_{\mathbf{M}}(\Omega)=\inf_{v\in\mathbf{M}(\Omega)}J(v), \] where $\mathbf{M}(\Omega)=\{ u\in X(\Omega)\backslash \{0\}: a(u)=b(u)\} $. Note that $\mathbf{M}(\Omega)$ contains every nonzero solution of \eqref{E1}. Consider the unit sphere $\mathbf{U}(\Omega)$ and the zero energy manifold $\mathbf{Z}(\Omega)$, where \begin{gather*} \mathbf{U}(\Omega)=\{u\in X(\Omega): \| u\| _{H^{1}}=1\},\\ \mathbf{Z}(\Omega)=\{u\in X(\Omega)\backslash\{0\}: \frac{1} {2}a(u)=\frac{1}{p}b(u)\}. \end{gather*} $\alpha_{\mathbf{M}}(\Omega)>0$ is a consequence of part $(i)$ of the following lemma. Part $(ii)$ of the following lemma will be used later in Lemma \ref{i7} and Theorem \ref{i13}. \begin{lemma} \label{i2} $(i)$ There is a bijective $C^{1,1}$ map $m$ from $\mathbf{U}(\Omega)$ to $\mathbf{M}(\Omega)$. Moreover, $\mathbf{M}(\Omega)$ is path-connected and a constant $c>0$ exists such that for $u\in\mathbf{M} (\Omega)$, $\| u\| _{H^{1}}\geq c$ and\ $J(u)\geq c$;\newline $(ii)$ There is a bijective $C^{1,1}$ map $z$ from $\mathbf{U} (\Omega)$ to $\mathbf{Z}(\Omega)$. Moreover, $\mathbf{Z}(\Omega)$ is path-connected and a constant $c'>0$ exists such that for $u\in\mathbf{Z}(\Omega)$, $\| u\| _{H^{1}}\geq c'$. \end{lemma} \begin{proof} $(i)$ For $t\geq0$, $u\in\mathbf{U}(\Omega)$, let \[ h_{u}(t)=J(tu) =\frac{1}{2}t^{2}-\frac{1}{p}t^{p}b(u). \] Then $h_{u}'(t)=t-t^{p-1}b(u)$. We take uniquely $s_{u}\in \mathbb{R}^{+}$ such that $s_{u}>0$, $s_{u}u\in\mathbf{M}(\Omega)$, and $0=h_{u}'(s_{u})$. For $v\in\mathbf{U}(\Omega)$, a $s_{v} \in\mathbb{R}^{+}$ exists such that $s_{v}v\in\mathbf{M}(\Omega)$: that is \[ \langle J'(s_{v}v) ,s_{v}v\rangle =s_{v}^{2}-s_{v}^{p}b(v)=0. \] Consider the function $g(t,u):\mathbb{R}^{+}\times \mathbf{U}(\Omega)\to\mathbb{R}$ defined by \[ g(t,u) =\langle J'(tu),tu\rangle =t^{2}a(u)-t^{p}b(u). \] Note that $g(s_{v},v) =\langle J'(s_{v}v) ,s_{v}v\rangle =0$. Thus, \[ \frac{\partial g}{\partial t}(t,u) \big| _{(s_{v},v) } =2s_{v}-ps_{v}^{p-1}b(v)=s_{v}(2-p)<0. \] By the implicit function theorem, a neighborhood $\mathbf{W}$ of $v$ in $\mathbf{U}(\Omega)$ and a unique function $t\in C^{1,1}$ exist such that \begin{gather*} t:\mathbf{W}\to\mathbb{R}^{+},\ t(v)=s_{v},\\ g(t(u) ,u) =0\;\text{for all }u\in\mathbf{W}. \end{gather*} Therefore, for each $v\in\mathbf{U}(\Omega)$, $t:\mathbf{U}(\Omega)\to\mathbb{R}^{+}$ and $m:\mathbf{U}(\Omega)\to \mathbf{M}(\Omega)$, $t$, $m\in C^{1,1}$ exist such that $t(v)=s_{v}$, $m(v)=s_{v}v$. Clearly, $t$ and $m$ are injective. For each $u\in \mathbf{M}(\Omega)$, write $u=s_{v}v$, where $s_{v}=\| u\|_{H^{1}}$ and $v=\frac{u}{\| u\| _{H^{1}}}\in\mathbf{U}(\Omega)$. Since $m(v)=u$, $m$ is surjective. Since $\mathbf{U}(\Omega)$ is path-connected, $\mathbf{M}(\Omega)$ is path-connected. Note that $u\in\mathbf{M}(\Omega)$, so $J'(u) =0$, or $s_{v} ^{2}=\int_{\Omega}s_{v}^{p}| v| ^{p}$. By the Sobolev embedding theorem, we have $s_{v}^{2}=\int_{\Omega}s_{v}^{p}| v| ^{p}\leq ds_{v}^{p}$, or $c\leq s_{v}$, where $d$ and $c$ are two positive constants. Therefore, $\|u\| _{H^{1}}=\| s_{v}v\| _{H^{1} }=s_{v}\geq c$ for $u\in\mathbf{M}(\Omega)$.\newline $(ii)$ The proof is similar to part $(i)$. \end{proof} \begin{theorem} \label{i3} Let $\beta>0$ and let $\{ u_{n}\} $ in $X(\Omega )\backslash\{0\}$ be a sequence for $J$ such that $J(u_{n})=\beta + o(1)$ and $a(u_{n})=b(u_{n})+ o(1)$. Then there is a sequence $\{s_{n}\}$ in $\mathbb{R}^{+}$ such that $s_{n}=1+ o(1)$, $\{s_{n}u_{n}\}$ is in $\mathbf{M}(\Omega) $ and $J(s_{n} u_{n})=\beta+ o(1)$. In particular, if $\{ u_{n}\} $ is a (PS)$_{\beta}$-sequence for $J$, then there is a sequence $\{s_{n}\}$ in $\mathbb{R}^{+}$ such that $\{s_{n}u_{n}\}$ is in $\mathbf{M}(\Omega) $ and there is also a (PS)$_{\beta}$-sequence in $X(\Omega)$ for $J$. \end{theorem} \begin{proof} By Lemma \ref{i2}, there is a sequence $\{s_{n}\}$ in $\mathbb{R}^{+}$ such that $\{s_{n}u_{n}\}$ is in $\mathbf{M}(\Omega) :$ $s_{n} ^{2}a(u_{n})=s_{n}^{p}b(u_{n})$ for each $n$, because $a(u_{n})=b(u_{n} )+ o(1)$ and $J(u_{n})=\beta+ o(1)$ imply $s_{n} =1+ o(1)$. Therefore, $J(s_{n}u_{n})=\beta+ o(1)$. The last part follows from Lemma \ref{p30}. \end{proof} A minimizing sequence $\{u_{n}\}$ in $\mathbf{M}(\Omega)$ of $\alpha_{\mathbf{M}}(\Omega)$ is a (PS)$_{\alpha_{\mathbf{M}}(\Omega)} $-sequence in $X(\Omega)$ for $J$ . \begin{theorem}\label{i4} Let $\{u_{n}\}$ be in $X(\Omega)$. Then $\{u_{n}\}$ is a (PS)$_{\alpha_{\mathbf{M}}(\Omega)}$-sequence for $J$ if and only if $J( u_{n}) =\alpha_{\mathbf{M}}(\Omega)+ o(1) $ and $a(u_{n}) =b(u_{n}) + o(1) $. In particular, every minimizing sequence $\{u_{n}\}$ in $\mathbf{M}(\Omega)$ of $\alpha_{\mathbf{M}}(\Omega)$ is a (PS)$_{\alpha_{\mathbf{M}}( \Omega) } $-sequence in $X(\Omega)$ for $J$ . In particular, $\alpha_{\mathbf{M}}(\Omega)$ is a (PS)$_{\alpha_{\mathbf{M}}(\Omega)}-$value in $X(\Omega)$ for $J$ . \end{theorem} \begin{proof} Suppose $\{ u_{n}\} $ is a (PS)$_{\alpha_{\mathbf{M}}(\Omega)} $-sequence in $X(\Omega)$ for $J$. By Lemma \ref{p30}, we have $a(u_{n}) =b(u_{n}) + o(1) $. Conversely, let $\{ u_{n}\} $ satisfy $J(u_{n}) =\alpha_{\mathbf{M}}(\Omega)+ o(1) $ and $a(u_{n}) =b(u_{n}) + o(1) $. Then we have \begin{equation} a(u_{n}) =\frac{2p}{p-2}\alpha_{\mathbf{M}}(\Omega)+o(1) \quad \text{as }n\to\infty.\label{4-1} \end{equation} For $n=1,2,\dots$, denote \begin{equation} l_{n}(\varphi)=\int_{\Omega}|u_{n}|^{p-2}u_{n}\varphi\quad \text{for }\varphi\in X(\Omega).\label{4-2} \end{equation} Let $\phi\in\mathbf{U}(\Omega)$. By Lemma \ref{i2}, $t>0$ exists such that $t\phi\in\mathbf{M}(\Omega):\| t\phi\| _{H^{1}}^{2}=\| t\phi\| _{L^{p}}^{p}$; we conclude that $t=\| \phi\| _{L^{p}}^{\frac{-p}{p-2}}$ and \[ \alpha_{\mathbf{M}(\Omega)}\leq(\frac{1}{2}-\frac{1}{p}) \| t\phi\| _{H^{1}}^{2}=\frac{p-2}{2p}t^{2}=\frac{p-2} {2p}\| \phi\| _{L^{p}}^{\frac{-2p}{p-2}}. \] Therefore, $\| \phi\| _{L^{p}}\leq(\frac{2p}{p-2} \alpha_{\mathbf{M}}(\Omega)) ^{\frac{2-p}{2p}}$. For each $n$, \begin{align*} |l_{n}(\phi)| & =\big|\int_{\Omega}|u_{n}|^{p-2}u_{n}\phi\big|\\ &\leq\Big(\int_{\Omega}|u_{n}|^{p}\Big) ^{\frac{p-1}{p}} \Big(\int_{\Omega}|\phi|^{p}\Big) ^{1/p}\\ & \leq(\frac{2p}{p-2}\alpha_{\mathbf{M}}(\Omega)) ^{\frac {p-1}{p}}(\frac{2p}{p-2}\alpha_{\mathbf{M}}(\Omega)) ^{\frac{2-p}{2p}}+ o(1)\\ & =\big(\frac{2p}{p-2}\alpha_{\mathbf{M}}(\Omega)\big) ^{1/2}+ o(1) \quad\text{as }n\to\infty, \end{align*} we have \begin{equation} \Vert l_{n}\Vert_{X^{-1}}\leq(\frac{2p}{p-2}\alpha_{\mathbf{M}} (\Omega)) ^{1/2}+ o(1)\quad\text{as }n\to\infty.\label{4-3} \end{equation} Furthermore, by \eqref{4-2}, we have \begin{equation} \label{4-4} \begin{aligned} l_{n}(\frac{u_{n}}{\Vert u_{n}\Vert_{H^{1}}}) & =\frac {\int_{\Omega}|u_{n}|^{p}}{\Vert u_{n}\Vert_{H^{1}}}\\ &=(b(u_{n}))^{1/2}+ o(1)\\ & =(\frac{2p}{p-2}\alpha_{\mathbf{M}}(\Omega)) ^{1/2}+ o(1) \quad \text{as }n\to\infty \end{aligned} \end{equation} By (\ref{4-3}) and (\ref{4-4}), we conclude that \[ \Vert l_{n}\Vert_{X^{-1}}=(\frac{2p}{p-2}\alpha_{\mathbf{M}} (\Omega)) ^{1/2}+ o(1)\quad\text{as }n\to\infty. \] By the Riesz representation theorem, for each $n$, $w_{n}\in X(\Omega)$ exists such that, for each $\varphi\in X(\Omega)$, \[ l_{n}(\varphi)=\langle w_{n},\varphi\rangle_{H^{1}}=\int_{\Omega}(\nabla w_{n}\cdot\nabla\varphi+w_{n}\varphi), \] and \begin{equation} \Vert w_{n}\Vert_{H^{1}}=\Vert l_{n}\Vert_{X^{-1}}=(\frac{2p} {p-2}\alpha_{\mathbf{M}}(\Omega)) ^{1/2}+ o(1).\label{4-5} \end{equation} Consequently, \begin{equation} \langle w_{n},u_{n}\rangle_{H^{1}}=l_{n}(u_{n}) =\int_{\Omega }|u_{n}|^{p}=\frac{2p}{p-2}\alpha_{\mathbf{M}}(\Omega)+ o(1).\label{4-6} \end{equation} By \eqref{4-1}, \eqref{4-5}, and \eqref{4-6}, we obtain \begin{align*} \Vert u_{n}-w_{n}\Vert_{H^{1}}^{2} & =\langle u_{n},u_{n}\rangle_{H^{1} }-2\langle u_{n},w_{n}\rangle_{H^{1}}+\langle w_{n},w_{n}\rangle_{H^{1}}\\ & =\Vert u_{n}\Vert_{H^{1}}^{2}-2\langle u_{n},w_{n}\rangle_{H^{1}}+\Vert w_{n}\Vert_{H^{1}}^{2}\\ & =\frac{2p}{p-2}\alpha_{\mathbf{M}}(\Omega)-2\frac{2p}{p-2}\alpha _{\mathbf{M}}(\Omega)+\frac{2p}{p-2}\alpha_{\mathbf{M}}(\Omega)+ o(1)\\ & = o(1)\quad\text{as }n\to\infty. \end{align*} For $\varphi\in\mathbf{U}(\Omega)$, we have \begin{align*} \langle J'(u_{n}),\varphi\rangle & = {\int_{\Omega}} (\nabla u_{n}\cdot\nabla\varphi+u_{n}\varphi)- {\int_{\Omega}} |u_{n}|^{p-2}u_{n}\varphi\\ & =\langle u_{n},\varphi\rangle_{H^{1}}-\langle w_{n},\varphi\rangle_{H^{1} }=\langle u_{n}-w_{n},\varphi\rangle_{H^{1}}, \end{align*} so \[ \| J'(u_{n})\| _{X^{-1}}\leq\Vert u_{n}-w_{n}\Vert _{H^{1}}= o(1). \] We conclude that $J'(u_{n})= o(1)$ strongly in $X^{-1}(\Omega)$ as $n\to\infty$. \end{proof} If $u$ achieves $\alpha_{\mathbf{M}}(\Omega)$, then $u$ is a nonzero solution of \eqref{E1}. \begin{theorem} \label{i6} Let $u\in\mathbf{M}(\Omega)$ be such that $J(u)=\min_{v\in \mathbf{M}(\Omega)}J(v)$. Then $u$ is a nonzero solution of \eqref{E1}. \end{theorem} \begin{proof} Set $g(v)=a(v)-b(v)$ for $v\in X(\Omega)$. Note that $\langle g^{\prime }(u),u\rangle =(2-p)a(u)\neq0$. Since the minimum of $J$ is achieved at $u$ and is constrained in $\mathbf{M}(\Omega)$, by the Lagrange multiplier theorem, $\lambda\in\mathbb{R}$ exists such that $J'(u)=\lambda g'(u)$ in $X(\Omega)$. Thus, \[ 0=\langle J'(u),u\rangle =\lambda\langle g^{\prime }(u),u\rangle , \] or $\lambda=0$. Thus, $J'(u)=0$. Hence, $u$ is a weak solution of \eqref{E1} such that $J(u)=\alpha_{\mathbf{M}}(\Omega)$. \end{proof} \noindent(C) Consider the mountain pass minimax problem \[ \alpha_{\Gamma}(\Omega)=\inf_{g\in\Gamma(\Omega)}\max_{t\in[0,1]}J(g(t)), \] where $e\neq0$, $J(e)=0$, and \[ \Gamma(\Omega)=\{g\in C([0,1],X(\Omega)): g(0)=0,g(1)=e\}. \] Then $\alpha_{\Gamma}(\Omega)>0$ is a consequence of the following lemma. \begin{lemma} \label{i7} A ball $B(0;r)$ in $X(\Omega)$, $c>0$, and $e\notin\overline {B(0;r)}$ exist such that $J(e)=0$ and $\min_{v\in\partial B(0;r)} J(v)\geq c$. \end{lemma} \begin{proof} By Lemma \ref{i2} $(ii)$, for each $u\in\mathbf{U}(\Omega)$, there is a $t>0$ such that $J(tu)=0$. Let $e=tu$, then $J(e)=0$. Since for each $v\in X(\Omega)\backslash\{0\}$ \[ J(v)=\frac{1}{2}a(v)-\frac{1}{p}b(v), \] by the Sobolev inequality, there is a constant $c_{1}>0$ such that $b(v)\leq c_{1}a(v)^{p/2}$, and we have \[ J(v)\geq a(v)\{ \frac{1}{2}-\frac{c_{1}}{p}a(v)^{\frac{p-2}{2}}\}. \] Take $r>0$ such that $e\notin\overline{B(0;r)}$ and $\frac{1}{2}-\frac{c_{1} }{p}r^{p-2}\geq\frac{1}{4}$, then for $\| v\| _{H^{1}}=r$, we have \[ J(v)\geq c, \] where $c=\frac{1}{4}r^{2}$. \end{proof} We require the following lemma. \begin{theorem}[Ekeland variational principle]\label{i8} Let $M$ be a complete metric space with metric $d$ and let $F:M\to\mathbb{R}\cup\{+\infty\}$ be lower semi-continuous, bounded from below, and $\not \equiv \infty$. Then for any $\varepsilon>0$ and $\lambda>0$, and any $u\in M$ with \[ F(u)\leq\inf_{M}F+\varepsilon, \] there is an element $v\in M$ such that \begin{gather*} F(v)\leq F(u),\\ d(u,v)\leq\frac{1}{\lambda},\\ F(w)+\varepsilon\lambda d(v,w)>F(v)\quad \text{for }w\neq v. \end{gather*} \end{theorem} \begin{proof} It is sufficient to prove our assertion for $\lambda=1$. The general case is obtained by replacing $d$ by an equivalent metric $\lambda d$. We define the relation on $M$: \[ w\leq v\Longleftrightarrow F(w) +\varepsilon d(v,w) \leq F(v) . \] It is easy to see that this relation define a partial ordering on $M$. We now construct inductively a sequence $\{ u_{m}\} $ as follows: $u_{0}=u$; also assuming that $u_{m}$ has been defined, we set \[ S_{n}=\{ w\in M\;|\text{\ }w\leq u_{n}\} \] and choose $u_{n+1}\in S_{n}$ so that \[ F(u_{n+1}) \leq\underset{S_{n}}{\inf}F+\frac{1}{n+1}. \] Since $u_{n+1}\leq u_{n}$, $S_{n+1}\subset S_{n}$, and by the lower semicontinuity of $F$, $S_{n}$ is closed. We now show that \textrm{diam} $S_{n}\to0$. Indeed, if $w\in S_{n+1}$, then $w\leq u_{n+1}\leq u_{n} $ and consequently, \[ \varepsilon d(w,u_{n+1}) \leq F(u_{n+1}) -F( w) \leq \inf_{S_{n}} F+\frac{1}{n+1}-\underset{S_{n}}{\inf }F=\frac{1}{n+1}. \] This estimate implies \[ \mathop{\rm diam}S_{n+1}\leq\frac{2}{\varepsilon(n+1) } \] and our claim follows. The fact that $M$ is complete implies that \[ \cap_{n\geq 0}S_{n}=\{ v\} \] for some $v\in M$. In particular, $v\in S_{0}$, that is, $v\leq u_{0}=u$. Hence, \[ F(v) \leq F(u) -\varepsilon d(u,v) \leq F(u) . \] Moreover, \[ d(u,v) \leq\varepsilon^{-1}(F(u) -F( v) ) \leq\varepsilon^{-1}\big(\inf_M F+\varepsilon-\inf_M F\big) =1. \] To complete the proof we must show $w\leq v$ implies $w=v$. If $w\leq v$, then $w\leq u_{n}$ for each integer $n\geq0$, that is $w\in \cap_{n\geq0} S_{n}=\{ v\} $. \end{proof} \begin{lemma} \label{i9} Let $\Gamma(\Omega)$ be the complete metric space with the usual $L^{\infty}$ distance $d$ and $J\in C^{1}(X(\Omega),\mathbb{R}) $. Then for each $\varepsilon>0$ and each $f\in\Gamma(\Omega)$ such that \begin{equation} \max_{s\in[0,1]} J(f(s)) \leq \alpha_{\Gamma}(\Omega)+\varepsilon,\label{4-7} \end{equation} $v\in X(\Omega)$ exists such that \begin{gather*} \alpha_{\Gamma}(\Omega)-\varepsilon\leq J(v)\leq\max_{s\in[0,1]} J(f(s) ) ,\\ \mathop{\rm dist}(v,f([0,1]) ) \leq\varepsilon^{1/2},\\ | J'(v) | \leq\varepsilon^{1/2}. \end{gather*} \end{lemma} \begin{proof} Without loss of generality, we can assume that \begin{equation} 0<\varepsilon<\alpha_{\Gamma}(\Omega).\label{4-8} \end{equation} Let $f\in\Gamma(\Omega)$ satisfy the condition (\ref{4-7}). We define the function $\Phi:\Gamma(\Omega)\to\mathbb{R}$ by \[ \Phi(g) =\max_{s\in[0,1]}J(g(s)) . \] Then $(i)$ $\Phi$ is bounded below: $\Phi(g)\geq\alpha_{\Gamma}(\Omega)>0$. \newline$(ii)$ $\Phi$ is continuous at each $g\in\Gamma(\Omega):$ since $J$ is continuous on the compact set $K=g([0,1]) $, for each $\varepsilon>0$, $u\in K$, there is a $\delta_{u}>0$ such that if $w\in B(u;\delta_{u})$ is an open ball in $X(\Omega)$, then $|J(w)-J(u)|<\frac{1} {2}\varepsilon$. Since $K$ is compact, finite values $B(u_{i};\delta_{u_{i}} )$, $i=1,\dots,n$, exist such that \[ K\subset B(u_{1};\frac{\delta_{u_{1}}}{2})\cup\dots\cup B(u_{n};\frac {\delta_{u_{n}}}{2}). \] Take $\delta=\min\{ \frac{\delta_{u_{1}}}{2},\dots,\frac{\delta_{u_{n} }}{2}\} $. Let $k\in\Gamma(\Omega)$ satisfy $\| k-g\| _{L^{\infty}}<\delta$. For each $s\in[0,1]$, we have \[ | k(s)-g(s)| <\delta, \] or $g(s)\in B(u_{i};\frac{\delta_{u_{i}}}{2})$, $k(s)\in B(u_{i};\delta _{u_{i}})$. Thus \[ |J(k(s))-J(g(s))|<\varepsilon, \quad\mbox{or}\quad | \Phi(k) -\Phi(g) | \leq \varepsilon. \] The Ekeland variational principle (Theorem \ref{i8}) implies the existence of $h\in\Gamma(\Omega)$ such that \begin{gather*} \Phi(h) \leq\Phi(f) \leq\alpha_{\Gamma}(\Omega)+\varepsilon,\\ \max_{s\in[0,1]}| h(s) -f(s)| \leq\varepsilon^{1/2}, \end{gather*} and \begin{equation} \Phi(g) >\Phi(h) -\varepsilon^{\frac{1}{2} }d(h,g) \quad\text{whenever }g\in\Gamma(\Omega)\quad\text{and }g\neq h.\label{4-9} \end{equation} Let $A=\{ s\in[0,1]:\alpha_{\Gamma}(\Omega)-\varepsilon\leq J( h(s) ) \} $, then $A$ is nonempty since \[ \alpha_{\Gamma}(\Omega)-\varepsilon<\alpha_{\Gamma}(\Omega)=\underset {g\in\Gamma(\Omega)}{\inf}\underset{s\in[0,1]}{\max}J(g(s))\leq\underset {s\in[0,1]}{\max}J(h(s)). \] Note that for $s\in A$, \[ | J'(h(s) ) | \leq \varepsilon^{1/2}, \] if and only if \[ | \langle J'(h(s) ) ,v\rangle | \leq\varepsilon^{1/2}\quad \text{for } v\in\mathbf{U}(\Omega), \] if and only if \[ \langle J'(h(s) ) ,v\rangle \geq-\varepsilon^{1/2}\quad \text{for }v\in\mathbf{U}(\Omega). \] We claim that there is some $s\in A$ satisfying $|J'(h(s) ) | \leq\varepsilon^{1/2}$. If this is not the case, then for each $s\in A$, $v_{s}\in\mathbf{U}(\Omega)$ exists such that $\langle J'(h(s) ) ,v_{s}\rangle <-\varepsilon^{1/2}$. By the continuity of $J'$, $\delta_{s}>0$ and an open ball $B_{s}$ in $[0,1]$ containing $s$ exist such that for $t\in B_{s}$ and $u\in X(\Omega)$ with $| u| \leq\delta_{s}$, we have \begin{equation} \langle J'(h(t) +u) ,v_{s} \rangle <-\varepsilon^{1/2}.\label{4-10} \end{equation} Since $A$ is compact, a finite subcovering $B_{s_{1}}$, $B_{s_{2}}$ \dots $B_{s_{k}}$ of $A$ exists. We define the Lipschitz continuous functions, for each $j=1,2,\dots,k$, $\psi_{j}:[0,1]\to[0,1]$ by \[ \psi_{j}(t) =\begin{cases} \mathop{\rm dist}(t,B_{s_{j}}^{c}) / \sum_{i=1}^k \mathop{\rm dist}(t,B_{s_{i}}^{c}) &\text{for }t\in A;\\ 0&\text{for }t\notin\cup_{i=1}^{k}B_{s_{i}}. \end{cases} \] Then \begin{gather*} \sum_{j=1}^k \psi_{j}(t) =1\text{for }t\in A;\\ \|\sum_{j=1}^k \psi_{j}(t) v_{s_{j}}\| _{H^{1}}\leq1\quad \text{for }t\in A. \end{gather*} Let $\delta=\min\{\delta_{s_{1}},\dots \delta_{s_{k}}\} $ and let $\psi:[0,1]\to[0,1] $ be a continuous function such that \[ \psi(t) =\begin{cases} 1 & \text{if }J(h(t) ) \geq\alpha_{\Gamma}(\Omega);\\ 0 & \text{if }J(h(t) ) \leq\alpha_{\Gamma}(\Omega)-\varepsilon, \end{cases} \] and let $g\in C([0,1],X(\Omega)) $ be defined by \[ g(t) =h(t) +\delta\psi(t) \overset{k}{\underset{j=1}{\sum}}\psi_{j}(t) v_{s_{j}}. \] It follows from (\ref{4-8}) that, for $t\in\{0,1\}$, we have $J( h(t) ) =0<\alpha_{\Gamma}(\Omega)-\varepsilon$, or $\psi(t) =0$. Consequently, $g(0) =h( 0) =0$ and $g(1) =h(1) =e$, that is, $g\in\Gamma(\Omega)$. The mean value theorem and (\ref{4-10}) imply that, for each $t\in A$, there is some $0<\tau<1$ for which \begin{equation} \begin{aligned} &J(g(t) ) -J(h(t) ) \\ &=\langle J'(h(t) +\tau\delta\psi\big( t) \overset{k}{\underset{j=1}{\sum}}\psi_{j}(t) v_{s_{j} }\big) ,\delta\psi(t) \overset{k}{\underset{j=1}{\sum}} \psi_{j}(t) v_{s_{j}}\rangle \\ & =\delta\psi(t) \overset{k}{\underset{j=1}{\sum}}\psi _{j}(t) \langle J'\big(h(t) +\tau\delta\psi(t) \overset{k}{\underset{j=1}{\sum}}\psi _{j}(t) v_{s_{j}}\big) ,v_{s_{j}}\rangle \\ & \leq-\varepsilon^{1/2}\delta\psi(t) . \end{aligned}\label{4-11} \end{equation} Thus \[ J(g(t) ) \leq J(h(t) ) -\varepsilon^{1/2}\delta\psi(t) \leq J(h( t) ) . \] If $t\notin A$, then $\psi(t) =0$ and hence $J(g( t) )=J(h(t) ) $. Let $\bar{t}\in[0,1]$ satisfy $J(g(\overline{t}) )=\Phi(g) $, then we obtain \[ J(h(\overline{t}) )\geq J(g(\overline{t}) )\geq\alpha_{\Gamma}(\Omega), \] so that $\overline{t}\in A$ and $\psi(\overline{t}) =1$. By (\ref{4-11}), we obtain \[ J(g(\overline{t}) )-J(h(\overline{t}) )\leq-\varepsilon^{1/2}\delta \] and in particular \[ \Phi(g) +\varepsilon^{1/2}\delta\leq J(h( \overline{t}) )\leq\Phi(h) , \] so that $g\neq h$. However, by the definition of $g$, we have $d(g,h) \leq\delta$ and \[ \Phi(g) +\varepsilon^{1/2}d(g,h) \leq \Phi(h) \] which contradicts (\ref{4-9}). The proof is complete . \end{proof} $\alpha_{\Gamma}(\Omega)$ is a (PS)-value in $X(\Omega)$ for $J$. \begin{theorem} \label{i10} Under the conditions of Lemma \ref{i9}, for each minimizing sequence $\{f_{k}\}\subset\Gamma(\Omega)$ such that \[ \Phi(f_{k}) =\underset{s\in[0,1]}{\max}J(f_{k}( s) ) =\alpha_{\Gamma}(\Omega)+ o(1), \] there is a (PS)-sequence $\{v_{k}\}$ in $X(\Omega)$ for $J$ satisfying \begin{gather*} J(v_{k})=\alpha_{\Gamma}(\Omega)+ o(1),\\ \mathop{\rm dist}(v_{k},\;f_{k}([0,1])) = o(1),\\ J'(v_{k})= o(1)\quad \text{strongly in }X^{-1}(\Omega) \end{gather*} as $k\to\infty$. In particular, $\alpha_{\Gamma}(\Omega)$ is a (PS)-value in $X(\Omega)$ for $J$. \end{theorem} \begin{proof} We define $\varepsilon_{k}=\underset{s\in[0,1]}{\max}J(f_{k}( s) ) -\alpha_{\Gamma}(\Omega)$ if $\underset{s\in[0,1]}{\max }J(f_{k}(s) ) -\alpha_{\Gamma}(\Omega)>0$ and $\varepsilon_{k}=\frac{1}{k}$ in the other case. Then we apply Lemma \ref{i9} to $\varepsilon_{k}$ and $f_{k}$: \begin{gather*} \alpha_{\Gamma}(\Omega)-\varepsilon_{k}\leq J(v_{k}) \leq\underset{s\in [0,1]}{\max}J(f_{k}(s) ) \leq\alpha_{\Gamma }(\Omega)+\varepsilon_{k},\\ \mathop{\rm dist}(v_{k},f_{k}([0,1]) ) \leq \varepsilon_{k}^{1/2},\\ | J'(v_{k}) | \leq\varepsilon_{k}^{\frac {1}{2}}\quad \quad\text{for each }k>0. \end{gather*} This completes the proof. \end{proof} \noindent (D) Consider the infimum of positive (PS)-values in $X(\Omega)$ for $J:$ \[ \alpha_{\mathbf{P}}(\Omega)=\inf_{\beta\in\mathbf{P}(\Omega) }\beta, \] where $\mathbf{P}(\Omega) $ is the set of all positive (PS)-values in $X(\Omega)$ for $J$. That $\alpha_{\mathbf{P}}(\Omega)>0$ is a consequence of the following theorem. \begin{theorem} \label{i11} There is a $\beta_{0}>0$ such that $\beta\geq\beta_{0}$ for every positive (PS)-value $\beta$ in $X(\Omega)$ for $J$. \end{theorem} \begin{proof} Let $\{ u_{n}\} $ be a (PS)$_{\beta}$-sequence in $X(\Omega)$ for $J$ with $\beta>0$. By Lemma \ref{p30}, a positive sequence $\{ c_{n}(\beta)\} $ exists such that $c_{n}(\beta)= o(1)$ as $n\to \infty$, $\beta\to0$, and \begin{equation} a(u_{n})\leq c_{n}(\beta)^{2}.\label{4-12} \end{equation} By the Sobolev embedding theorem, there is a constant $d>0$ such that \begin{equation} b(u_{n})\leq da(u_{n})^{p/2}.\label{4-13} \end{equation} By Lemma \ref{p30}, (\ref{4-12}), and (\ref{4-13}), we have \[ o(1) =a(u_{n})-b(u_{n}) \geq a(u_{n})\left[ 1-dc_{n}(\beta)^{p-2}\right] . \] Take $\beta_{0}>0$ and $n_{0}>0$ such that if $\beta<\beta_{0}$ and $n\geq n_{0}$, then $1-dc_{n}(\beta)^{p-2}>\frac{1}{2}$. Consequently, $a(u_{n} )=b(u_{n})= o(1)$, or $J(u_{n})= o(1)$. Thus, $\beta\geq\beta_{0}$. \end{proof} $\alpha_{\mathbf{P}}(\Omega) $ is a (PS)-value in $X(\Omega) $ for $J$. \begin{theorem} \label{i12} $\alpha_{\mathbf{P}}(\Omega)\in\mathbf{P}(\Omega)$. \end{theorem} \begin{proof} For each $n\in\mathbb{N}$, take $u_{n}\in X(\Omega)$ and $c_{n}\in \mathbf{P}(\Omega)$ such that \begin{gather*} |c_{n}-\alpha_{\mathbf{P}}(\Omega)|<\frac{1}{n},\\ |J(u_{n})-c_{n}|<\frac{1}{n},\\ \| J'(u_{n})\| _{X^{-1}}<\frac{1}{n}. \end{gather*} Then $J(u_{n})=\alpha_{\mathbf{P}}(\Omega)+ o(1)$ and $J'(u_{n})= o(1)$. Thus, $\alpha_{\mathbf{P}}(\Omega)\in\mathbf{P}(\Omega)$. \end{proof} The following theorem is very useful. \begin{theorem} \label{i13} Let $\beta$ be a positive (PS)-value in $X(\Omega)$ for $J$. Then \newline$(i)\;\beta\geq\alpha_{\gamma}(\Omega);\;\;(ii)\;\beta\geq \alpha_{\mathbf{M}}(\Omega);\;\;(iii)\;\beta\geq\alpha_{\Gamma}(\Omega)$; $(iv) $ $\beta\geq\alpha_{\mathbf{P}}(\Omega)$. \end{theorem} \begin{proof} Let $\{ u_{n}\} $ be a nonzero (PS)$_{\beta}$-sequence in $X(\Omega)$ for $J$ with $\beta>0$. By Lemma \ref{p30}, we have \begin{gather*} J(u_{n})=\beta+ o(1),\\ a(u_{n})-b(u_{n})= o(1). \end{gather*} $(i)$ Let $w_{n}=u_{n}(a(u_{n})) ^{-\frac{1}{2}}$, then $a(w_{n})=1$ and $b(w_{n})=a(u_{n})^{-p/2}b(u_{n})\leq\gamma(\Omega)^{p}$. Thus, $a(u_{n})\geq\gamma(\Omega)^{2p/(2-p)}+ o(1)$, or $\beta \geq(\frac{1}{2}-\frac{1}{p})\gamma(\Omega)^{2p/(2-p)}=\alpha_{\gamma} (\Omega)$.\newline $(ii)$ By Theorem \ref{i3}, there is a sequence $\{s_{n}\}$ in $\mathbb{R}^{+}$ such that $\{s_{n}u_{n}\}\subset\mathbf{M}(\Omega)$ and $J(s_{n}u_{n})=\beta+ o(1)$. Therefore, $\beta\geq\alpha_{\mathbf{M} }(\Omega)$.\newline $(iii)$ By Theorem \ref{i3} and Lemma \ref{i2} $(ii) $, there are sequences $\{s_{n}\}$ and $\{t_{n}\}$ in $\mathbb{R}^{+}$ such that $\{s_{n}u_{n}\}\subset\mathbf{M}(\Omega)$, $\{t_{n}u_{n}\}\subset\mathbf{Z}(\Omega)$, and $J(s_{n}u_{n})=\beta + o(1)$. Since the manifold $\mathbf{Z}(\Omega)$ is path-connected, there is a path $\zeta_{n}$ in $\mathbf{Z}(\Omega)$ that connects $t_{n}u_{n}$ to $e$. Let $\gamma_{n}'$ be the line segment connecting $0$ and $t_{n}u_{n}$ and the path $\gamma_{n}=\gamma_{n}'\cup\zeta_{n}$, then \[ \alpha_{\Gamma}(\Omega)\leq\underset{0\leq t\leq1}{\max}J(\gamma _{n}(t))=J(s_{n}u_{n}) =\beta+ o(1). \] Thus, $\beta\geq\alpha_{\Gamma}(\Omega)$.\newline $(iv)$ Clearly, $\beta \geq\alpha_{\mathbf{P}}(\Omega)$. \end{proof} By Theorems \ref{i1}, \ref{i4}, \ref{i10}, \ref{i12}, and \ref{i13}, we have the following theorem. \begin{theorem} \label{i14} $\alpha_{\gamma}(\Omega)=\alpha_{\mathbf{M}}(\Omega)=\alpha _{\Gamma}(\Omega)=\alpha_{\mathbf{P}}(\Omega)$. \end{theorem} \begin{definition}\label{i15} \rm By Theorem \ref{i14}, we conclude that the positive (PS)-values $\alpha_{\gamma}(\Omega)$, $\alpha_{\Gamma}(\Omega)$, $\alpha_{\mathbf{M} }(\Omega)$, and$\;\alpha_{\mathbf{P}}(\Omega)$ in $X(\Omega)$ for $J$ are the same. Any one of them is called the index of $J$ in $X(\Omega) $ and denoted by $\alpha_{X}(\Omega)$. By the definition of $\alpha_{\mathbf{M} }(\Omega)$, if $u$ is a nonzero solution of Equation $($\ref{E1}$)$, then $u\in\mathbf{M}(\Omega)$. Thus, $J(u)\geq\alpha_{\mathbf{M}}(\Omega )=\alpha_{X}(\Omega)$. We say that a nonzero solution $u $ of Equation \eqref{E1} in $X(\Omega) $ is a ground state solution if $J(u)=\alpha_{X}(\Omega)$, and is a higher energy solution if $J(u)>\alpha _{X}(\Omega)$. \end{definition} \begin{remark} \rm We denote $\alpha_{X}(\Omega)$ by $\alpha(\Omega)$ for $X( \Omega) =H_{0}^{1}(\Omega) $ and by $\alpha_{s}(\Omega)$ for $X(\Omega) =H_{s}^{1}(\Omega) $ (see Definition \ref{w1}). \end{remark} \begin{remark}\label{b201} \rm By Theorem \ref{b2}, a ground state solution in $X( \Omega) $ is of constant sign. Note that if $u$ is a solution of \eqref{E1}, then $-u$ is also a solution of \eqref{E1}. By the maximum principle, if $u$ is a nonzero and nonnegative solution of \eqref{E1}, then $u$ is positive. From now on, by a ground state solution in $X(\Omega) $, we mean a positive solution of \eqref{E1}. \end{remark} \begin{definition} \rm We say that a domain $\Omega$ in $\mathbb{R}^{N}$ is an achieved domain if there is a solution $u$ in $H_{0}^{1}(\Omega)$ of \eqref{E1} such that $J(u)=\alpha(\Omega)$, by Remark \ref{b201}, we may assume that $u$ be positive. Otherwise, we say that $\Omega$ is a nonachieved domain. \end{definition} \begin{theorem}\label{f8} $(i)$ If $\Omega$ is a large domain in $\mathbb{R}^{N}$, then $\alpha(\Omega)=\alpha(\mathbb{R}^{N})$; \newline $(ii)$ If $\Omega$ is a large domain in $\mathbf{A}^{r}$, then $\alpha(\Omega)=\alpha(\mathbf{A}^{r} )$; \newline $(iii)$ If $\Omega$ is a large domain in $\mathbf{A}^{r_{1},r_{2}} $, then $\alpha(\Omega)=\alpha(\mathbf{A}^{r_{1},r_{2}})$. \end{theorem} \begin{proof} It suffices to prove part $(i)$. Let $w\in H^{1}(\mathbb{R}^{N})$ be a ground state solution of Equation \eqref{E1} satisfying \[ a(w)=\int_{\mathbb{R}^{N}}(|\nabla w|^{2}+w^{2})=b(w)=\int_{\mathbb{R}^{N} }|w|^{p}=(\frac{2p}{p-2}) \alpha(\mathbb{R}^{N}). \] For $r_{n}\to\infty$, take $\{z_{n}\}\subset\Omega$ such that $B^{N}(z_{n};r_{n})\subset\Omega$. Consider the cut-off function $\eta\in C_{c}^{\infty}([ 0,\infty) ) $ as in \eqref{1-2}, and for each $n$, let \[ w_{n}(z)=\eta(\frac{2|z-z_{n}|}{r_{n}})w(z-z_{n}). \] Then $w_{n}\in H_{0}^{1}(\Omega)$ and \begin{gather*} a(w_{n})={\int_{\Omega}} (|\nabla w_{n}|^{2}+w_{n}^{2})=(\frac{2p}{p-2}) \alpha (\mathbb{R}^{N})+ o(1),\\ b(w_{n})={\int_{\Omega}} |w_{n}|^{p}=(\frac{2p}{p-2}) \alpha(\mathbb{R}^{N})+ o(1)\quad\text{ as }n\to\infty. \end{gather*} Thus, \begin{gather*} J(w_{n}) =\alpha(\mathbb{R}^{N})+ o(1) ,\\ a(w_{n})=b(w_{n})+ o(1) \quad\text{as }n\to\infty. \end{gather*} By Theorem \ref{i4}, $\{ w_{n}\} $ is a (PS)$_{\alpha (\mathbb{R}^{N})}$-sequence in $H_{0}^{1}(\Omega)$ for $J$. Therefore, $\alpha(\Omega)\leq\alpha(\mathbb{R}^{N})$. Clearly, $\alpha(\mathbb{R} ^{N})\leq\alpha(\Omega)$, thus we have $\alpha(\Omega)=\alpha(\mathbb{R}^{N})$. \end{proof} \begin{theorem} \label{f9} Let $\Omega$ be a large domain in $\mathbb{R}^{N}$. If $\beta$ is a positive (PS)-value in $H_{0}^{1}(\Omega)$ for $J$, then $m\beta$ is also a positive (PS)-value in $H_{0}^{1}(\Omega)$ for $J$, where $m=2,3,\dots$. \end{theorem} \begin{proof} It suffices to prove the case $m=2$. First embed $H_{0}^{1}(\Omega)$ into $H^{1}(\mathbb{R}^{N})$. Let $\{ u_{n}\} $ be a (PS)$_{\beta} $-sequence in $H_{0}^{1}(\Omega)$. Then by Lemma \ref{p30}, there is a constant $c$ $>0$ such that, for each $n$, $a(u_{n}) \leq c$ and $b(u_{n}) \leq c$. For $r_{n}\to\infty$, since $\Omega\backslash B^{N}(0;5r_{n})$ is also a large domain in $\mathbb{R}^{N}$, $z_{n}\in\Omega$ exists such that $B^{N}(z_{n};2r_{n})\subset\Omega$ and \[ {\int_{B^{N}(0;r_{n})^{c}}} | \nabla u_{n}| ^{2}+u_{n}^{2}<\frac{1}{n}\quad\text{and}\quad {\int_{B^{N}(0;r_{n})^{c}}}| u_{n}| ^{p}<\frac{1}{n}. \] Note that $|z_{n}|\geq5r_{n}$. Let $\eta_{n}(z)=\eta(\frac{|z|}{r_{n}}) $, where $\eta$ is as in (\ref{1-2}), $v_{n}(z)=\eta_{n}(z)u_{n}(z)$ and $w_{n}(z)=v_{n}(z-z_{n})$. Then we have $|\nabla\eta_{n}|\leq\frac{2}{r_{n}}$ and $\mathop{\rm supp} w_{n}\subset B^{N}(z_{n};2r_{n}) $.\newline $(i)$ $J(v_{n})=\beta+ o(1)$: note that \[ |\nabla v_{n}|^{2} =|\eta_{n}|^{2}|\nabla u_{n}|^{2}+|\nabla\eta_{n} |^{2}|u_{n}|^{2}+2\eta_{n}u_{n}\nabla\eta_{n}\nabla u_{n}. \] Thus, for $z\in B^{N}(0;r_{n})$, we have $|\nabla v_{n}|=|\nabla u_{n}|$ and \begin{align*} {\int_{\Omega}} |\nabla v_{n}|^{2} & = {\int_{B^{N}(0;r_{n})}} |\nabla v_{n}|^{2}+ {\int_{B^{N}(0;2r_{n})\backslash B^{N}(0;r_{n})}} |\nabla v_{n}|^{2}\\ & = {\int_{B^{N}(0;r_{n})}} |\nabla u_{n}|^{2}+ o(1)\\ & = {\int_{\Omega}} |\nabla u_{n}|^{2}+ o(1). \end{align*} Similarly, we have \[ {\int_{\Omega}}|v_{n}|^{2}= {\int_{\Omega}}|u_{n}|^{2}+ o(1),\quad {\int_{\Omega}} |v_{n}|^{p}= {\int_{\Omega}} |u_{n}|^{p}+ o(1). \] Thus, $J(v_{n})=J(u_{n})+ o(1)=\beta+ o(1)$. Clearly, for each $n$, $J(w_{n})=J(v_{n})$, and hence $J(w_{n})=\beta+ o(1)$.\newline $(ii)$ $J(v_{n}+w_{n})=2\beta+ o(1):$ since the supports of $v_{n}$ and $w_{n}$ are disjoint, we have \begin{align*} a(v_{n}+w_{n}) & ={\int_{\Omega}} | \nabla(v_{n}+w_{n}) | ^{2}+(v_{n} +w_{n}) ^{2}\\ & ={\int_{\Omega}} | \nabla v_{n}| ^{2}+v_{n}^{2}+ {\int_{\Omega}} | \nabla w_{n}| ^{2}+w_{n}^{2} +2{\int_{\Omega}} \nabla v_{n}\nabla w_{n}+2 {\int_{\Omega}} v_{n}w_{n}\\ & =a(v_{n})+a(w_{n}). \end{align*} Now, \begin{align*} &{\int_{\Omega}} | v_{n}+w_{n}| ^{p}-| v_{n}| ^{p}-|w_{n}| ^{p}\\ & ={\int_{B^{N}(0;2r_{n})}} | v_{n}+w_{n}| ^{p}-| v_{n}| ^{p}-|w_{n}| ^{p} + {\int_{B^{N}(0;2r_{n})^{c}\cap\Omega}} | v_{n}+w_{n}| ^{p}-| v_{n}| ^{p}-|w_{n}| ^{p}\\ & =0. \end{align*} Thus, \[ b(v_{n}+w_{n}) ={\int_{\Omega}}| v_{n}+w_{n}| ^{p} ={\int_{\Omega}}| v_{n}| ^{p}+ {\int_{\Omega}}| w_{n}| ^{p}\\ =b(v_{n})+b(w_{n}). \] Hence, \[ J(v_{n}+w_{n}) =\frac{1}{2}a(v_{n}+w_{n})-\frac{1}{p}b(v_{n}+w_{n}) =J(v_{n})+J(w_{n}) =2\beta+ o(1). \] $(iii)$ $\| J'(v_{n}+w_{n})\| = o(1):$ for $\varphi\in C_{c}^{\infty}(\Omega)$, we have \begin{align*} \langle J'(v_{n}) ,\varphi\rangle & =\int_{B^{N}(0;r_{n})}u_{n}(\nabla\eta_{n})\cdot\nabla\varphi+\int _{B^{N}(0;r_{n})}\eta_{n}(\nabla u_{n})\cdot\nabla\varphi+\eta_{n}u_{n} \varphi\\ & -\int_{B^{N}(0;r_{n})}| \eta_{n}u_{n}| ^{p-2}\eta_{n} u_{n}\varphi+ o(1)\\ & =\int_{B^{N}(0;r_{n})}\nabla u_{n}(z)\cdot\nabla\varphi(z)+u_{n} (z)\varphi(z)\\ &\quad -\int_{B^{N}(0;r_{n})}| u_{n}| ^{p-2}u_{n}\varphi(z)+ o(1)\\ & =\langle J'(u_{n}) ,\varphi\rangle + o(1). \end{align*} Thus, $\| J'(v_{n}) \| _{H^{-1}}= o(1)$. Similarly, $\| J'(w_{n}) \| _{H^{-1} }= o(1)$.\newline We have \begin{align*} &{\int_{\Omega}} | v_{n}+w_{n}| ^{p-2}(v_{n}+w_{n}) \varphi-| v_{n}| ^{p-2}v_{n}\varphi-| w_{n}| ^{p-2}w_{n}\varphi\\ & ={\int_{B^{N}(0;2r_{n})}} | v_{n}+w_{n}| ^{p-2}(v_{n}+w_{n}) \varphi-| v_{n}| ^{p-2}v_{n}\varphi-| w_{n}| ^{p-2}w_{n}\varphi\\ & \quad+ {\int_{B^{N}(0;2r_{n})^{c}\cap\Omega}} | v_{n}+w_{n}| ^{p-2}(v_{n}+w_{n}) \varphi-| v_{n}| ^{p-2}v_{n}\varphi-| w_{n}| ^{p-2}w_{n}\varphi\\ & =0. \end{align*} Now for $\varphi\in C_{c}^{\infty}(\Omega) $, we have \begin{align*} \langle J'(v_{n}+w_{n}) ,\varphi\rangle & = {\int_{\Omega}} \nabla(v_{n}+w_{n}) \nabla\varphi+(v_{n}+w_{n}) \varphi\\ &\quad -{\int_{\Omega}} | v_{n}+w_{n}| ^{p-2}(v_{n}+w_{n})\varphi\\ & ={\int_{\Omega}} \nabla v_{n}\nabla\varphi+v_{n}\varphi+ {\int_{\Omega}} \nabla w_{n}\nabla\varphi+w_{n}\varphi\\ &\quad -{\int_{\Omega}} | v_{n}| ^{p-2}v_{n}\varphi- {\int_{\Omega}} | w_{n}| ^{p-2}w_{n}\varphi\\ & =\langle J'(v_{n}),\varphi\rangle +\langle J'(w_{n}),\varphi\rangle . \end{align*} Therefore, $\| J'(v_{n}+w_{n}) \| _{H^{-1} }= o(1)$. We conclude that \begin{gather*} J(v_{n}+w_{n})=2\beta+ o(1),\\ J'(v_{n}+w_{n})= o(1)\quad \quad\text{strongly in }H^{-1}(\Omega). \end{gather*} \end{proof} The following theorem has a proof similar to that of Theorem \ref{f9}. \begin{theorem} \label{f91} Let $\Omega$ be a large domain in $\mathbf{A}^{r}$. If $\beta$ is a positive (PS)-value in $H_{0}^{1}(\Omega)$ for $J$, then $m\beta$ is also a positive (PS)-value in $H_{0}^{1}(\Omega)$ for $J$, where $m=2,3,\dots$. \end{theorem} \begin{lemma} \label{f10} The set $\mathbf{P}(\Omega) $ is closed. \end{lemma} The proof of this lemma is similar to the proof of Theorem \ref{i12}; so we omit it. By Lemma \ref{i2}, $J(\mathbf{M}(\Omega)) $ is bounded below away from zero. Actually for any domain $\Omega$ in $\mathbb{R}^{N}$, $J(\mathbf{M}(\Omega)) $ is unbounded above. \begin{theorem} \label{f14} If $\Omega$ is a domain in $\mathbb{R}^{N}$, then $J(\mathbf{M}(\Omega)) =(\alpha(\Omega),\infty) $ for a nonachieved domain $\Omega$ and $J(\mathbf{M}(\Omega)) =[ \alpha(\Omega),\infty) $ for an achieved domain $\Omega$. \end{theorem} \begin{proof} $(i)$ Suppose that $\Omega$ is bounded. By Struwe \cite[p.116 Theorem 6.6]{S}, an unbounded sequence $\{ u_{n}\} $ exists in $\mathbf{M}(\Omega)$ for $J$. Since $J(u_{n}) =(\frac{1}{2}-\frac{1}{p}) a(u_{n})$ and $\mathbf{M}(\Omega)$ is path connected, then we have $J(\mathbf{M}(\Omega)) =[\alpha(\Omega),\infty)$. \newline $(ii)$ Let $\Omega$ be an unbounded domain and $\Omega_{1}$ be a bounded domain in $\Omega$. Then $\mathbf{M}( \Omega_{1}\mathbf{)\subset M(}\Omega)$ and $\alpha( \Omega\mathbf{)\leq}\alpha(\Omega_{1})$. By part $(i)$, we have \[ [ \alpha(\Omega_{1}),\infty) =J(\mathbf{M}(\Omega_{1})) \subset J(\mathbf{M}(\Omega)) . \] Since $\mathbf{M}(\Omega)$ is path connected, the result follows. \end{proof} \begin{theorem} \label{f15} Let $\Omega$ be an Esteban-Lions domain as well as a large domain in $\mathbb{R}^{N}$. Then we have $\mathbf{P}(\Omega) =\{ \alpha(\Omega) ,\;2\alpha(\Omega) ,\;3\alpha(\Omega) ,\dots\} $. \end{theorem} \begin{proof} By Theorem \ref{f9}, $\mathbf{P}(\Omega) \supset\{ \alpha(\Omega) ,\;2\alpha(\Omega) ,\;3\alpha (\Omega) ,\dots\} $. Suppose that a (PS)$_{\beta} $-sequence $\{ u_{n}\} $ exists for $J$, where $k\alpha( \Omega) <\beta<(k+1) \alpha(\Omega) $ for some $k\in\mathbb{N}$. By the Palais-Smale decomposition theorem \ref{d1}, Equation \eqref{E1} has a nonzero solution. This contradicts Theorem \ref{n3}. \end{proof} By Lemma \ref{p30}, if $\{ u_{n}\} $ is a (PS)$_{\beta}- $sequence in $H_{0}^{1}(\Omega)$ for $J$, then $a(u_{n})=b(u_{n} )+ o(1)=\frac{2p}{p-2}\beta+ o(1)$. By Theorems \ref{f14} and \ref{f15}, we have: \begin{lemma} \label{f16} Let $\Omega$ be an Esteban-Lions domain as well as a large domain in $\mathbb{R}^{N}$. For each $\beta$ and $m=0,1,\dots$, satisfying $m\alpha(\Omega) <\beta<(m+1)\alpha(\Omega) $, then there is a sequence $\{ u_{n}\} $ in $H_{0}^{1}(\Omega)$ for $J\ $satisfying \[ a(u_{n})=b(u_{n})+ o(1)=\frac{2p}{p-2}\beta+ o(1) \] but \[ J'(u_{n})\nrightarrow0\quad\text{strongly in }H^{-1}(\Omega). \] \end{lemma} Let $\Omega$ be an unbounded domain in $\mathbb{R}^{N}$ and $\Omega_{n} =\Omega\cap\mathbf{B}^{N}(0;r_{n}) $, then we have the following theorem. \begin{theorem} \label{f177} $\alpha_{X}(\Omega_{n}) =\alpha_{X}( \Omega) + o(1)$ as $n\to\infty$. \end{theorem} \begin{proof} Suppose that $\{ u_{n}\} $ in $X(\Omega) $ is a minimizing sequence in $\mathbf{M}(\Omega) $ of $\alpha _{X}(\Omega) $, then by Lemma \ref{p30}, $\{ u_{n}\} $ is bounded in $X(\Omega) $. Let $\{r_{n}\} $ be a sequence of strictly increasing positive integers such that $r_{n}\geq n$, \begin{equation} \int_{\Omega\cap\{ | z| \geq\frac{r_{n}}{2}\} }| \nabla u_{n}| ^{2}+| u_{n}| ^{2}<\frac{1} {n}\label{5-1} \end{equation} and \begin{equation} \int_{\Omega\cap\{ | z| \geq\frac{r_{n}}{2}\} }| u_{n}| ^{p}<\frac{1}{n}.\label{5-2} \end{equation} Define $\eta_{n}(z) =\eta(\frac{2| z| }{r_{n}}) $, where $\eta$ is as in $( \ref{1-2}) $. Then $\eta_{n}u_{n}\in X( \Omega_{n}) \subset X(\Omega) $. From the inequalities $(\ref{5-1})$ and $(\ref{5-2})$, we obtain \[ a(\eta_{n}u_{n})=a(u_{n})+ o(1)\quad\text{and }b(\eta_{n}u_{n})=b(u_{n} )+ o(1). \] Therefore, we have \[ J(\eta_{n}u_{n}) =J(u_{n}) + o(1) =\alpha_{X}(\Omega) + o(1). \] and \[ a(\eta_{n}u_{n})=b(\eta_{n}u_{n})+ o(1). \] By Theorem \ref{i3}, there is a sequence $\{ s_{n}\} $ in $\mathbb{R}^{+}$ such that $s_{n}=1+ o(1)$, $\{ s_{n}\eta_{n} u_{n}\} $ is in $\mathbf{M}(\Omega)$ and $J(s_{n}\eta_{n}u_{n} )=\alpha_{X}(\Omega) + o(1)$. Note that $J(s_{n}\eta _{n}u_{n}) \geq\alpha_{X}(\Omega_{n}) >\alpha_{X} (\Omega)$. Hence, $\alpha_{X}(\Omega_{n}) =\alpha_{X}( \Omega) + o(1)$. \end{proof} Let $\Omega$ be a domain containing zero in $\mathbb{R}^{N}$. For $\delta>0$, we define \[ \delta\Omega=\{ \delta z\mid z\in\Omega\} . \] Then we have the following theorem. \begin{theorem} \label{f178} $(i)$ $\lim_{\delta\to\infty}\al