\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2014 (2014), No. 82, pp. 1--14.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2014 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2014/82\hfil Multiple positive solutions] {Multiple positive solutions for a critical elliptic problem with concave and convex nonlinearities} \author[H. Fan \hfil EJDE-2014/82\hfilneg] {Haining Fan} % in alphabetical order \address{Haining Fan \newline College of Sciences, China University of Mining and Technology, Xuzhou 221116, China} \email{fanhaining888@163.com} \thanks{Submitted January 18, 2013. Published March 26, 2014.} \subjclass[2000]{35J20, 58J05} \keywords{Nehari manifold; critical Sobolev exponent; positive solution; \hfill\break\indent semi-linear elliptic problem; Ljusternik-Schnirelmann category} \begin{abstract} In this article, we study the multiplicity of positive solutions for a semi-linear elliptic problem involving critical Sobolev exponent and concave-convex nonlinearities. With the help of Nehari manifold and Ljusternik-Schnirelmann category, we prove that problem admits at least $\operatorname{cat}(\Omega)+1$ positive solutions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction and main result} Let us consider the semi-linear problem \begin{equation} \label{eElambda} \begin{gathered} -\Delta u=\lambda|u|^{q-2}u+|u|^{2^*-2}u, \quad x\in \Omega,\\ u>0, \quad x\in \Omega,\\ u=0, \quad x\in \partial\Omega, \end{gathered} \end{equation} where $\Omega$ is an open bounded domain in $\mathbb{R}^N$ with smooth boundary, $10, \quad x\in \Omega,\\ u=0,\quad x\in \partial\Omega. \end{gathered} \end{equation} It is well known that the existence of solutions of \eqref{eEbar} is affected by the shape of the domain $\Omega$. This has been the focus of a great deal of research by several authors. In particular, the first striking result is due to Pohozaev \cite{p1} who proved that if $\Omega$ is star-shaped with respect to some point, \eqref{eEbar} has no solution. However, if $\Omega$ is an annulus, Kazdan and Warner \cite{k2} pointed out that \eqref{eEbar} has at least one solution. For a non-contractible domain $\Omega$, Coron \cite{c1} proved that \eqref{eEbar} has a solution. Further existence results for ``rich topology" domain, we refer to \cite{a2,k1,k2,o1,p1,s1,t1,t2}. The fact that the number of solutions of \eqref{eElambda} is affected by the concave-convex nonlinearities and the domain $\Omega$ has been the focus of a great deal of research in recent years. In particular, Ambrosetti, Brezis and Cerami \cite{a3} showed that there exists $\lambda_0>0$ such that \eqref{eElambda} admits at least two solutions for $\lambda\in (0,\lambda_0)$, one solution for $\lambda=\lambda_0$ and no solution for $\lambda>\lambda_0$. Actually, Adimurthi et al. \cite{a5}, Ouyang and Shi \cite{o1} and Tang \cite{t2} proved that there exists $\lambda_0>0$ such that \eqref{eElambda} in unit ball $B^N(0;1)$ has exactly two solutions for all $\lambda\in(0,\lambda_0)$, exactly one solution for $\lambda=\lambda_0$ and no solution for all $\lambda>\lambda_0$. Recently, when $\Omega$ is a non-contractible domain, Wu \cite{w1} showed that \eqref{eElambda} admits at least three solutions if $\lambda$ is small enough. In this work we aim to get a better information on the number of solutions of \eqref{eElambda}, for small value of parameter $\lambda$, via the Nehari manifold and Ljusternik-Schnirelmann category. Our main result is as follows. \begin{theorem} \label{thm1.1} There exists $\lambda_0>0$ such that, for each $\lambda\in(0,\lambda_0)$, problem \eqref{eElambda} has at least $\operatorname{cat}(\Omega)+1$ solutions. \end{theorem} Here $\operatorname{cat}$ means the Ljusternik-Schnirelmann category and for properties of it we refer to Struwe \cite{s1}. \begin{remark} \label{rmk1.1} \rm If $\Omega$ is a general domain, $\operatorname{cat}(\Omega)\geq1$ and Theorem \ref{thm1.1} is the result of \cite{a3}. If $\Omega$ is non-contractible, $\operatorname{cat}(\Omega)\geq2$ and Theorem \ref{thm1.1} is the result of Wu \cite{w1}. \end{remark} Associated with \eqref{eElambda}, we consider the energy functional $J_\lambda$ for each $H_0^1(\Omega)$, \[ J_\lambda(u)=\frac{1}{2}\int_\Omega|\nabla u|^2dx -\frac{\lambda}{q}\int_\Omega (u_+)^qdx-\frac{1}{p^*}\int_\Omega(u_+)^{2^*}dx, \] where $u_+=\max\{u,0\}$. From the assumption, it is easy to prove that $J_\lambda$ is well defined in $H_0^1(\Omega)$ and $J_\lambda\in C^2(H_0^1(\Omega),\mathbb{R})$. Furthermore, the critical points of $J_\lambda$ are weak solutions of \eqref{eElambda}. We consider the behaviors of $J_\lambda$ on the Nehari manifold \[ S_\lambda=\{u\in H_0^1(\Omega)\setminus\{0\};u_+\not\equiv0\text{ and } \langle J'_\lambda(u),u\rangle=0\}, \] where $\langle,\rangle$ denotes the usual duality between $H_0^1(\Omega)$ and $H^{-1}(\Omega)$. This enables us to construct homotopies between $\Omega$ and certain levels of $J_\lambda$. Clearly, $u\in S_\lambda$ if and only if \[ \int_\Omega|\nabla u|^2dx-\lambda\int_\Omega(u_+)^qdx-\int_\Omega(u_+)^{2^*}dx=0. \] On the Nehari manifold $S_\lambda$, from the Sobolev embedding theorem and the Young inequality, we have \begin{equation} \label{e1.1} \begin{aligned} J_\lambda(u) &=\big(\frac{1}{2}-\frac{1}{2^*}\big)\int_\Omega|\nabla u|^2dx -\lambda\big(\frac{1}{q}-\frac{1}{2^*}\big)\int_\Omega(u_+)^qdx \\ &\geq\big(\frac{1}{2}-\frac{1}{2^*}\big)\int_\Omega|\nabla u|^2dx -\lambda\big(\frac{1}{q}-\frac{1}{2^*}\big) S_q^{-q}\Big(\int_\Omega|\nabla u|^2dx\Big)^{q/2} \\ &\geq\big(\frac{1}{2}-\frac{1}{2^*}\big)\int_\Omega|\nabla u|^2dx -\big(\frac{1}{2}-\frac{1}{2^*}\big)\int_\Omega|\nabla u|^2dx -D\lambda^{\frac{2}{2-q}}, \end{aligned} \end{equation} where $S_q$ is the best Sobolev constant for the embedding of $H_0^1(\Omega)$ into $L^q(\Omega)$ and $D$ is a positive constant depending on $q$ and $S_q$. Thus $J_\lambda$ is coercive and bounded below on $S_\lambda$. It is useful to understand $S_\lambda$ in terms of the fibrering maps $\phi_u(t)=J_\lambda(tu)(t>0)$. It is clear that, if $u\in S_\lambda$, then $\phi_u$ has a critical point at $t=1$. Furthermore, we will discuss the essential nature of $\phi_u$ in Section 2. This article is organized as follows: In Section 2, we give some notations and preliminary results. In Section 3, we discuss some concentration behavior. In Section 4, we give the proof of the main theorem. \section{Preliminaries} Throughout the paper by $|\cdot|_r$ we denote the $L^r$-norm. On the space $H_0^1(\Omega)$ we consider the norm \[ \|u\|=\Big(\int_\Omega|\nabla u|^2dx\Big)^{1/2}. \] Set also \[ \mathcal{D}^{1,2}(\mathbb{R}^N) :=\big\{u\in L^{2^*}(\mathbb{R}^N);\frac{\partial u}{\partial x_i} \in L^2(\mathbb{R}^N) \text{ for }i=1,\ldots,N\big\} \] equipped with the norm \[ \|u\|_*=\Big(\int_{\mathbb{R}^N}|\nabla u|^2dx\Big)^{1/2}. \] We will denote by $S$ the best Sobolev constant of the embedding $H_0^1(\Omega)\hookrightarrow L^{2^*}(\Omega)$ given by \[ S:=\inf\big\{\int_\Omega|\nabla u|^2dx;u\in H_0^1(\Omega), |u|_{2^*}=1\big\}. \] It is known that $S$ is independent of $\Omega$ and is never achieved except when $\Omega=\mathbb{R}^N$ (see \cite{t1}). We then define the Palais-Smale(simply by $(PS)$) sequences, $(PS)$-values, and $(PS)$-conditions in $H_0^1(\Omega)$ for $J_\lambda$ as follows. \begin{definition} \rm \begin{itemize} \item[(i)] For $\beta\in\mathbb{R}$, a sequence $\{u_k\}$ is a $(PS)_\beta$-sequence in $H_0^1(\Omega)$ for $J_\lambda$ if $J_\lambda(u_k)=\beta+o(1)$ and $J'_\lambda(u_k)=o(1)$ strongly in $H^{-1}(\Omega)$ as $k\to\infty$. \item[(ii)] $J_\lambda$ satisfies the $(PS)_\beta$-condition in $H_0^1(\Omega)$ if every $(PS)_\beta$-sequence in $H_0^1(\Omega)$ for $J_\lambda$ contains a convergent subsequence. \end{itemize} \end{definition} We now define \begin{equation} \label{e2.1} \psi_\lambda(u):=\langle J'_\lambda(u),u\rangle =\int_\Omega|\nabla u|^2dx-\lambda\int_\Omega (u_+)^qdx-\int_\Omega(u_+)^{2^*}dx. \end{equation} Then for $u\in S_\lambda$, \begin{align} \langle \psi'_\lambda(u),u\rangle &=(2-q)\int_\Omega|\nabla u|^2dx-(2^*-q)\int_\Omega(u_+)^{2^*}dx \label{e2.2} \\ &=(2-2^*)\int_\Omega|\nabla u|^2dx+\lambda(2^*-q)\int_\Omega(u_+)^qdx. \label{e2.3} \end{align} Similarly to the method used in \cite{b1}, we split $S_\lambda$ into three parts: \begin{gather*} S_\lambda^+ =\{u\in S_\lambda;\langle\psi_\lambda'(u),u\rangle>0\}, \\ S_\lambda^0 =\{u\in S_\lambda;\langle\psi_\lambda'(u),u\rangle=0\}, \\ S_\lambda^- =\{u\in S_\lambda;\langle\psi_\lambda'(u),u\rangle<0\}. \end{gather*} Then we have the following results. \begin{lemma} \label{lem2.1} Suppose that $u_0$ is a local minimum for $J_\lambda$ on $S_\lambda$. Then, if $u_0\not\in S_\lambda^0$, $u_0$ is a critical point of $J_\lambda$. \end{lemma} \begin{proof} Since $u_0$ is a local minimum for $J_\lambda$ on $S_\lambda$, then $u_0$ is a solution of the optimization problem \[ \text{minimize $J_\lambda(u)$ subject to $\psi_\lambda(u)=0$.} \] Hence, by the theory of Lagrange multipliers, there exists $\mu\in\mathbb{R}$ such that $J'_\lambda(u_0)=\mu\psi_\lambda'(u_0)$ in $H^{-1}(\Omega)$. Thus, \begin{equation} \label{e2.4} \langle J'_\lambda(u_0),u_0\rangle=\mu\langle\psi_\lambda'(u_0),u_0\rangle. \end{equation} Since $u_0\in S_\lambda$, we obtain $\langle J'_\lambda(u_0),u_0\rangle=0$. However, $u_0\not\in S_\lambda^0$ and so by \eqref{e2.4} $\mu=0$ and $J'_\lambda(u_0)=0$. This completes the proof. \end{proof} \begin{lemma} \label{lem2.2} There exists $\lambda_1>0$ such that for each $\lambda\in(0,\lambda_1)$, we have $S_\lambda^0=\emptyset$. \end{lemma} \begin{proof} Suppose otherwise, that is $S_\lambda^0\neq\emptyset$ for all $\lambda>0$. Then for $u\in S_\lambda^0$, we from \eqref{e2.2}, \eqref{e2.3} and the Sobolev embedding theorem obtain that there are two positive numbers $c_1$, $c_2$ independent of $u$ and $\lambda$ such that \[ \int_\Omega|\nabla u|^2dx \leq c_1\Big(\int_\Omega|\nabla u|^2dx\Big)^{2^*/2},\quad \int_\Omega|\nabla u|^2dx\leq \lambda c_2 \Big(\int_\Omega|\nabla u|^2dx\Big)^{q/2} \] or \[ \int_\Omega|\nabla u|^2dx\geq c_1^{-\frac{2}{2^*-2}},\quad \int_\Omega|\nabla u|^2dx\leq (\lambda c_2)^{\frac{2}{2-q}}. \] If $\lambda$ is sufficiently small, this is impossible. Thus we can conclude that there exists $\lambda_1>0$ such that for each $\lambda\in(0,\lambda_1)$, we have $S_\lambda^0=\emptyset$. \end{proof} By Lemma \ref{lem2.2}, for $\lambda\in(0,\lambda_1)$, we write $S_\lambda=S_\lambda^+\cup S_\lambda^-$ and define \[ \alpha_\lambda^+=\inf_{u\in S_\lambda^+}J_\lambda(u),\quad \alpha_\lambda^-=\inf_{u\in S_\lambda^-}J_\lambda(u). \] We now discuss the nature of the fibrering maps $\phi_u(t)$. It is useful to consider the function \begin{equation} \label{e2.5} M_u(t)=t^{2-q}\int_\Omega|\nabla u|^2dx-t^{2^*-q}\int_\Omega(u_+)^{2^*}dx. \end{equation} Clearly, for $t>0$, $tu\in S_\lambda$ if and only if $t$ is a solution of \begin{equation} \label{e2.6} M_u(t)=\lambda\int_\Omega(u_+)^qdx. \end{equation} Moreover, we have from $M'_u(t)=0$ know that there is a unique critical point $t_{\rm max}$: \[ t_{\rm max}=\bigg(\frac{(2-q)\int_\Omega|\nabla u|^2dx}{(2^*-q) \int_\Omega(u_+)^{2^*}dx}\bigg)^{1/(2^*-2)}. \] Furthermore, the direct computation gives that \[ M''_u(t_{\rm max})=(2^*-q)(2-p^*)t_{\rm max}^{2^*-q-2}\int_\Omega(u_+)^{2^*}dx<0. \] This shows that $M_u(t)$ is increasing in $(0,t_{\rm max})$ and decreasing for $t\geq t_{\rm max}$. Suppose $tu\in S_\lambda$. It follows from \eqref{e2.2} and \eqref{e2.5} that if $M'_u(t)>0$, then $tu\in S_\lambda^+$, and if $M'_u(t)<0$, then $tu\in S_\lambda^-$. If $\lambda>0$ is sufficiently large, \eqref{e2.6} has no solution and so $\phi_u(t)$ has no critical point, in this case $\phi_u(t)$ is a decreasing function. Hence no multiple of $u$ lies in $S_\lambda$. If, on the other hand, $\lambda>0$ is sufficiently small, there are exactly two solutions $t_1(u)0$ and $M'_u(t_2(u))<0$. Thus there are exactly two multiples of $u\in S_\lambda$, that is, $t_1(u)u\in S_\lambda^+$ and $t_2(u)u\in S_\lambda^-$. It follows that $\phi_u(t)$ has exactly two critical points, a local minimum at $t_1(u)$ and a local maximum at $t_2(u)$. Moreover, $\phi_u(t)$ is decreasing in $(0,t_1(u))$, increasing in $(t_1(u),t_2(u))$ and decreasing in $(t_2(u),\infty)$. Then we have the following result. \begin{lemma} \label{lem2.3} \begin{itemize} \item[(i)] $\alpha_\lambda^+<0$. \item[(ii)] There exist $\lambda_2, \delta>0$ such that $\alpha_\lambda^-\geq\delta$ for all $\lambda\in(0,\lambda_2)$. \end{itemize} \end{lemma} \begin{proof} (i) Given $u\in S_\lambda^+$, from \eqref{e2.3} and the definition of $S_\lambda^+$, we obtain \begin{align*} J_\lambda(u) &=\big(\frac{1}{2}-\frac{1}{2^*}\big)\int_\Omega|\nabla u|^2dx -\lambda\big(\frac{1}{q}-\frac{1}{2^*}\big)\int_\Omega(u_+)^qdx\\ &\leq\big[\big(\frac{1}{2}-\frac{1}{2^*}\big)-\big(\frac{1}{q} -\frac{1}{2^*}\big)\frac{2^*-2}{2^*-q}\big]\int_\Omega|\nabla u|^2dx\\ &=\frac{2^*-2}{2^*}\big(\frac{1}{2}-\frac{1}{q}\big)\int_\Omega|\nabla u|^2dx<0. \end{align*} This yields $\alpha_\lambda^+<0$. (ii) For $u\in S_\lambda^-$, by \eqref{e2.2} and the Sobolev embedding theorem, we obtain \begin{align*} (2-q)\int_\Omega|\nabla u|^2dx&<(2^*-q)\int_\Omega(u_+)^{2^*}dx\\ &\leq(2^*-q)S^{-\frac{2^*}{2}}\Big(\int_\Omega|\nabla u|^2dx\Big)^{2^*/2}. \end{align*} Thus there exists $c>0$ such that \[ \int_\Omega|\nabla u|^2dx\geq c. \] Moreover, \begin{align*} J_\lambda(u) &=\big(\frac{1}{2}-\frac{1}{2^*}\big)\int_\Omega|\nabla u|^2dx -\lambda\big(\frac{1}{q}-\frac{1}{2^*}\big)\int_\Omega(u_+)^qdx\\ &\geq\big(\frac{1}{2}-\frac{1}{2^*}\big)\int_\Omega|\nabla u|^2dx -\lambda\big(\frac{1}{q}-\frac{1}{2^*}\big)S_q^{-q} \Big(\int_\Omega|\nabla u|^2dx\Big)^{q/2}\\ &=\Big(\int_\Omega|\nabla u|^2dx\Big)^{q/2} \Big[\big(\frac{1}{2}-\frac{1}{2^*}\big) \Big(\int_\Omega|\nabla u|^2dx\Big)^{1-\frac{q}{2}} -\lambda\big(\frac{1}{q}-\frac{1}{2^*}\big)S_q^{-q}\Big]. \end{align*} Hence, there exist $\lambda_2, \delta>0$ such that $\alpha_\lambda^-\geq\delta$ for all $\lambda\in(0,\lambda_2)$. \end{proof} We establish that $J_\lambda$ satisfies the $(PS)_\beta$-condition under some condition on the level of $(PS)_\beta$-sequences in the following. \begin{lemma} \label{lem2.4} For each $\lambda\in(0,\lambda_2)$, $J_\lambda$ satisfies the $(PS)_\beta$-condition with $\beta$ in $(-\infty,\alpha_\lambda^++\frac{1}{N}S^{N/2})$. \end{lemma} \begin{proof} Let $\{u_k\}\subset H_0^1(\Omega)$ be a $(PS)_\beta$-sequence for $J_\lambda$ and $\beta\in(-\infty,\alpha_\lambda^++\frac{1}{N}S^{N/2})$. After a standard argument(see \cite{w2}), we know that $\{u_k\}$ is bounded in $H_0^1(\Omega)$. Thus, there exists a subsequence still denoted by $\{u_k\}$ and $u\in H_0^1(\Omega)$ such that $u_k\rightharpoonup u$ weakly in $H_0^1(\Omega)$. By the compactness of Sobolev embedding and the Brezis-Lieb Lemma \cite{w2}, we obtain \begin{gather*} \lambda\int_\Omega(u_k)_+^qdx=\lambda\int_\Omega (u_+)^qdx+o(1),\\ \int_\Omega|\nabla u_k-\nabla u|^2dx=\int_\Omega|\nabla u_k|^2dx -\int_\Omega|\nabla u|^2dx+o(1),\\ \int_\Omega(u_k-u)_+^{2^*}dx=\int_\Omega(u_k)_+^{2^*}dx -\int_\Omega(u_+)^{2^*}dx+o(1). \end{gather*} Moreover, we can obtain $J'_\lambda(u)=0$ in $H^{-1}(\Omega)$. Since $J_\lambda(u_k)=\beta+o(1)$ and $J'_\lambda(u_k)=o(1)$ in $H^{-1}(\Omega)$, we deduce that \begin{equation} \label{e2.7} \frac{1}{2}\int_\Omega|\nabla u_k-\nabla u|^2dx -\frac{1}{2^*}\int_\Omega(u_k-u)_+^{2^*}dx=\beta-J_\lambda(u)+o(1) \end{equation} and \[ \int_\Omega|\nabla u_k-\nabla u|^2dx-\int_\Omega(u_k-u)_+^{2^*}dx=o(1). \] Now we may assume that \[ \int_\Omega|\nabla u_k-\nabla u|^2dx\to l, \quad \int_\Omega(u_k-u)_+^{2^*}dx\to l\quad \text{as }k\to\infty, \] for some $l\in[0,+\infty)$. Suppose $l\neq0$. Using the Sobolev embedding theorem and passing to the limit as $k\to\infty$, we have $l\geq Sl^{2/2^*}$; that is, \begin{equation} \label{e2.8} l\geq S^{N/2}. \end{equation} Then by \eqref{e2.7}, \eqref{e2.8} and $u\in S_\lambda$, we have \[ \beta=J_\lambda(u)+\frac{1}{N}l\geq\frac{1}{N}S^{N/2}+\alpha_\lambda^+, \] which contradicts the definition of $\beta$. Hence $l=0$, that is, $u_k\to u$ strongly in $H_0^1(\Omega)$. \end{proof} Then we obtain the following result. \begin{lemma} \label{lem2.5} For each $0<\lambda<\min\{\lambda_1,\lambda_2\}$, the functional $J_\lambda$ has a minimizer $u_\lambda^+$ in $S_\lambda^+$ and it satisfies: \begin{itemize} \item[(i)] $J_\lambda(u_\lambda^+)=\alpha_\lambda^+ =\inf_{u\in S_\lambda^+}J_\lambda(u)$; \item[(ii)] $u_\lambda^+$ is a solution of \eqref{eElambda}; \item[(iii)] $J_\lambda(u_\lambda^+)\to0$ as $\lambda\to0$. \item[(iv)] $\lim_{\lambda\to0}\|u_\lambda^+\|=0$. \end{itemize} \end{lemma} \begin{proof} (i)--(iii) are consequences in \cite[Theorem 1.1]{k1}. Now we show (iv). By (i)--(iii), we have \begin{equation} \label{e2.9} 0=\lim_{\lambda\to0}J_\lambda(u_\lambda^+) =\lim_{\lambda\to0}\Big(\frac{1}{N}\int_\Omega|\nabla u_\lambda^+|^2dx -\big(\frac{1}{q}-\frac{1}{2^*}\big)\lambda\int_\Omega(u_\lambda^+)^qdx\Big). \end{equation} Since $J_\lambda$ is coercive and bounded below on $S_\lambda$, $\int_\Omega|\nabla u_\lambda^+|^2dx$ is bounded and so that \begin{equation} \label{e2.10} \lim_{\lambda\to0}\lambda\int_\Omega(u_\lambda^+)^qdx=0. \end{equation} Hence, from \eqref{e2.9} and \eqref{e2.10} we complete the proof. \end{proof} \section{Concentration behavior} In this Section, we will recall and prove some Lemmas which are crucial in the proof of the main theorem. Firstly, we denote $c_\lambda:=\frac{1}{N}S^{N/2}+\alpha_\lambda^+$ and consider the filtration of the manifold $S_\lambda^-$ as follows: \[ S_\lambda^-(c_\lambda):=\{u\in S_\lambda^-;J_\lambda(u)\leq c_\lambda\}. \] In Section 4, we will prove that \eqref{eElambda} admits at least $\operatorname{cat}(\Omega)$ solutions in this set. Then we need the following Lemmas. \begin{lemma} \label{lem3.1} Let $\{u_k\}\subset H_0^1(\Omega)$ be a nonnegative function sequence with $|u_k|_{2^*}=1$ and $\int_\Omega|\nabla u_k|^2dx\to S$. Then there exists a sequence $(y_k,\lambda_k)\in\mathbb{R}^N\times\mathbb{R}^+$ such that \[ \upsilon_k(x):=\lambda_k^{\frac{N-2}{2}}u_k(\lambda_kx+y_k) \] contains a convergent subsequence denoted again by $\{\upsilon_k\}$ such that $\upsilon_k\to\upsilon$ in $\mathcal{D}^{1,2}(\mathbb{R}^N)$ with $\upsilon(x)>0$ in $\mathbb{R}^N$. Moreover, we have $\lambda_k\to0$ and $y_k\to y\in\overline{\Omega}$. \end{lemma} For a proof of the above lemma, see Willem \cite{w2}. \begin{lemma} \label{lem3.2} Suppose that $X$ is a Hilbert manifold and $F\in C^1(X,\mathbb{R})$. Assume that for $c_0\in\mathbb{R}$ and $k\in\mathbb{N}$: \begin{itemize} \item[(i)] $F(x)$ satisfies the $(PS)_c$ condition for $c\leq c_0$, \item[(ii)] $\operatorname{cat}(\{x\in X;F(x)\leq c_0\})\geq k$. \end{itemize} Then $F(x)$ has at least $k$ critical points in $\{x\in X;F(x)\leq c_0\}$. \end{lemma} For a proof of the above lemma, see See \cite[Theorem 2.3]{a1}. Up to translations, we may assume that $0\in\Omega$. Moreover, in what follows, we fix $r>0$ such that $B_r=\{x\in\mathbb{R}^N;|x|r\} \] are both homotopically equivalent to $\Omega$. Now we define the continuous map $\Phi:S_\lambda^-\to\mathbb{R}^N$ by setting \[ \Phi(u):=\frac{\int_\Omega x(u_+)^{2^*}dx}{\int_\Omega(u_+)^{2^*}dx}. \] \begin{lemma} \label{lem3.3} There exists $\lambda_3>0$ such that if $\lambda\in(0,\lambda_3)$ and $u\in S_\lambda^-(c_\lambda)$, then $\Phi(u)\in\Omega_r^+$. \end{lemma} \begin{proof} By way of contradiction, let $\{\lambda_k\}$ and $\{u_k\}$ be such that $\lambda_k\to0$, $u_k\in S_{\lambda_k}^-(c_{\lambda_k})$ and $\Phi(u_k)\not\in\Omega_r^+$. From \eqref{e1.1}, we have that $\{u_k\}$ is bounded in $H_0^1(\Omega)$ and $\lambda_k\int_\Omega(u_k)_+^qdx\to0$. Thus, by Lemma \ref{lem2.5} (iii) we have \begin{equation} \label{e3.1} \lim_{k\to\infty}J_{\lambda_k}(u_k) =\lim_{k\to\infty}\frac{1}{N}\int_\Omega|\nabla u_k|^2dx =\lim_{k\to\infty}\frac{1}{N}\int_\Omega(u_k)_+^{2^*}dx\leq\frac{1}{N}S^{N/2}. \end{equation} Defining $\omega_k=u_k/|(u_k)_+|_{2^*}$, we see that $|(\omega_k)_+|_{2^*}=1$. By \eqref{e3.1} and the definition of $S$, we obtain \[ \lim_{k\to\infty}\int_\Omega|\nabla\omega_k|^2dx =\lim_{k\to\infty}\int_\Omega|\nabla(\omega_k)_+|^2dx=S. \] Furthermore, the functions $\widetilde{\omega}_k=(\omega_k)_+$ satisfy \begin{equation} \label{e3.2} |\widetilde{\omega}_k|_{2^*}=1,\quad \int_\Omega|\nabla\widetilde{\omega}_k|^2dx\to S. \end{equation} By Lemma \ref{lem3.1}, there is $\{\varepsilon_k\}$ in $\mathbb{R}^+$ and $\{y_k\}$ in $\mathbb{R}^N$, such that $\varepsilon_k\to0$, $y_k\to y\in\overline{\Omega}$ and $\upsilon_k(x) =\varepsilon_k^{\frac{N-2}{N}}\widetilde{\omega}_k(\varepsilon_kx+y_k)\to\upsilon$ in $\mathcal{D}^{1,2}(\mathbb{R}^N)$ with $\upsilon(x)>0$ in $\mathbb{R}^N$. Considering $\varphi\in C_0^\infty(\mathbb{R}^N)$ such that $\varphi(x)=x$ in $\Omega$, we infer \begin{align} \Phi(u_k)=\frac{\int_\Omega x(u_k)_+^{2^*}dx}{\int_\Omega (u_k)_+^{2^*}dx}&=\int_{\mathbb{R}^N}\varphi(x)(\widetilde{\omega}_k)^{2^*}dx =\int_{\mathbb{R}^N}\varphi(\varepsilon_kx+y_k)(\upsilon_k(x))^{2^*}dx. \end{align} Moreover, by Lebesgue Theorem, \[ \int_{\mathbb{R}^N}\varphi(\varepsilon_kx+y_k)(\upsilon_k(x))^{2^*}dx\to y \in\overline{\Omega}, \] so that $\lim_{k\to\infty}\Phi(u_k)=y\in\overline{\Omega}$, in contradiction with $\Phi(u_k)\not\in\Omega_r^+$. \end{proof} It is well known that $S$ is attained when $\Omega=\mathbb{R}^N$ by the functions \[ y_\varepsilon(x)=\frac{[N(N-2)\varepsilon^2]^{(N-2)/4}} {(\varepsilon^2+|x|^2)^{(N-2)/2}}. \] for any $\varepsilon>0$. Moreover, the functions $y_\varepsilon(x)$ are the only positive radial solutions of \[ -\Delta u=|u|^{2^*-2}u \] in $\mathbb{R}^N$. Hence, \[ S\Big(\int_{\mathbb{R}^N}|y_\varepsilon|^{2^*}dx\Big)^{2/2^*} =\int_{\mathbb{R}^N}|\nabla y_\varepsilon|^2dx =\int_{\mathbb{R}^N}|y_\varepsilon|^{2^*}dx=S^{N/2}. \] Let $0\leq\phi(x)\leq1$ be a function in $C_0^\infty(\Omega)$ defined as \[ \phi(x)=\begin{cases} 1, &\text{if } |x|\leq r/4,\\ 0, &\text{if } |x|\geq r/2. \end{cases} \] Assume \[ \upsilon_\varepsilon(x)=\phi(x)y_\varepsilon(x). \] The argument in \cite{s1} gives \begin{equation} \label{e3.3} \int_\Omega|\nabla\upsilon_\varepsilon|^2dx=S^{N/2}+O(\varepsilon^{N-2}),\quad \int_\Omega|\upsilon_\varepsilon|^{2^*}dx=S^{N/2}+O(\varepsilon^N). \end{equation} Moreover, we have the following result. \begin{lemma} \label{lem3.4} There exist $\varepsilon_0,\sigma(\varepsilon)>0$ such that for $\varepsilon\in(0,\varepsilon_0)$ and $\sigma\in(0,\sigma(\varepsilon))$, we have \[ \sup_{t\geq0}J_\lambda(u_\lambda^++t\upsilon_\varepsilon(x-y)) 0$ such that \[ u_\lambda^++t_{(\lambda,\varepsilon,y)}^-\upsilon_\varepsilon(x-y) \in S_\lambda^-(c_\lambda-\sigma),\quad \Phi(u_\lambda^++t_{(\lambda,\varepsilon,y)}^-\upsilon_\varepsilon(x-y)) \in\Omega_r^+. \] \end{lemma} \begin{proof} From Lemma \ref{lem2.5} and the definition of $\Omega_r^-$, we can define \begin{equation} \label{e3.4} c_0:=\inf_{M_r}u_\lambda^+>0, \end{equation} where $M_r:=\{x\in\Omega; \operatorname{dist}(x,\Omega_r^-)\leq\frac{r}{2}\}$. Since \begin{equation} \label{e3.5} \begin{aligned} &J_\lambda(u_\lambda^++t\upsilon_\varepsilon(x-y))\\ &=\frac{1}{2}\int_\Omega|\nabla(u_\lambda^++t\upsilon_\varepsilon(x-y))|^2dx -\frac{\lambda}{q}\int_\Omega|u_\lambda^++t\upsilon_\varepsilon(x-y)|^qdx\\ &\quad -\frac{1}{2^*}\int_\Omega|u_\lambda^++t\upsilon_\varepsilon(x-y)|^{2^*}dx\\ &=\frac{1}{2}\int_\Omega|\nabla u_\lambda^+|^2dx +\frac{t^2}{2}\int_\Omega|\nabla\upsilon_\varepsilon|^2dx+\langle u_\lambda^+, t\upsilon_\varepsilon(x-y)\rangle\\ &\quad -\frac{\lambda}{q}\int_\Omega|u_\lambda^++t\upsilon_\varepsilon(x-y)|^qdx -\frac{1}{2^*}\int_\Omega|u_\lambda^++t\upsilon_\varepsilon(x-y)|^{2^*}dx. \end{aligned} \end{equation} Note \eqref{e3.4} and a useful estimate obtained by Brezis and Nirenberg (see \cite[(17) and (21)]{b2}) shows that \begin{align*} &\int_\Omega|u_\lambda^++t\upsilon_\varepsilon(x-y)|^{2^*}dx\\ &=\int_\Omega|u_\lambda^+|^{2^*}dx+t^{2^*}\int_\Omega|\upsilon_\varepsilon|^{2^*}dx +2^*t\int_\Omega(u_\lambda^+)^{2^*-1}\upsilon_\varepsilon(x-y)dx \\ &\quad +2^*t^{2^*-1}\int_\Omega(\upsilon_\varepsilon(x-y))^{2^*-1}u_\lambda^+dx +o(\varepsilon^{\frac{N-2}{2}}) , \end{align*} uniformly in $y\in\Omega_r^-$. Substituting in \eqref{e3.5} and by Lemma \ref{lem2.5}, \eqref{e3.3}, \eqref{e3.4}, we obtain \begin{align*} &J_\lambda(u_\lambda^++t\upsilon_\varepsilon(x-y))\\ &=\frac{1}{2}\int_\Omega|\nabla u_\lambda^+|^2dx +\frac{t^2}{2}S^\frac{N}{2}+t\langle u_\lambda^+,\upsilon_\varepsilon(x-y)\rangle \\ &\quad -\frac{1}{2^*}\int_\Omega|u_\lambda^+|^{2^*}dx-\frac{t^{2^*}}{2^*} S^\frac{N}{2}-t\int_\Omega(u_\lambda^+)^{2^*-1}\upsilon_\varepsilon(x-y)dx \\ &\quad -t^{2^*-1}\int_\Omega(\upsilon_\varepsilon(x-y))^{2^*-1}u_\lambda^+dx -\frac{\lambda}{q}\int_\Omega|u_\lambda^++t\upsilon_\varepsilon(x-y)|^qdx +o(\varepsilon^{\frac{N-2}{2}})\\ &=J_\lambda(u_\lambda^+)+\frac{t^2}{2}S^\frac{N}{2}-\frac{t^{2^*}}{2^*}S^\frac{N}{2} -t^{2^*-1}\int_\Omega(\upsilon_\varepsilon(x-y))^{2^*-1}u_\lambda^+dx \\ &\quad -\frac{\lambda}{q}\int_\Omega|u_\lambda^++t\upsilon_\varepsilon(x-y)|^qdx +\frac{\lambda}{q}\int_\Omega|u_\lambda^+|^qdx \\ &\quad +t\lambda\int_\Omega(u_\lambda^+)^{q-1}\upsilon_\varepsilon(x-y)dx +o(\varepsilon^{\frac{N-2}{2}}) \\ &=\alpha_\lambda^++\frac{t^2}{2}S^\frac{N}{2}-\frac{t^{2^*}}{2^*}S^\frac{N}{2} -t^{2^*-1}\int_\Omega(\upsilon_\varepsilon(x-y))^{2^*-1}u_\lambda^+dx \\ &\quad -\lambda\int_\Omega\Big\{\int_0^{t\upsilon_\varepsilon(x-y)} [(u_\lambda^++s)^{q-1}-(u_\lambda^+)^{q-1}]ds\Big\}dx +o(\varepsilon^{\frac{N-2}{2}}) \\ &\leq\alpha_\lambda^++\frac{t^2}{2}S^\frac{N}{2}-\frac{t^{2^*}}{2^*} S^\frac{N}{2}-t^{2^*-1}\int_\Omega(\upsilon_\varepsilon(x-y))^{2^*-1} u_\lambda^+dx+o(\varepsilon^{\frac{N-2}{2}}) \end{align*} for all $y\in\Omega_r^-$. Applying \eqref{e3.4} and the fact that $\int_\Omega(\upsilon_\varepsilon(x-y))^{2^*-1}dx=O(\varepsilon^{\frac{N-2}{2}})$, also note the compactness of $\Omega_r^-$, we conclude that there exist $\varepsilon_0,\sigma(\varepsilon)>0$ such that for $\varepsilon\in(0,\varepsilon_0)$ and $\sigma\in(0,\sigma(\varepsilon))$, \begin{equation} \label{e3.6} \sup_{t\geq0}J_\lambda(u_\lambda^++t\upsilon_\varepsilon(x-y)) <\frac{1}{N}S^{N/2}+\alpha_\lambda^+-\sigma\quad \text{uniformly in } y\in\Omega_r^-. \end{equation} Next we will prove that there exists $t_{(\lambda,\varepsilon,y)}^->0$ such that $u_\lambda^++t_{(\lambda,\varepsilon,y)}^-\upsilon_\varepsilon(x-y)\in S_\lambda^-$ for each $y\in\Omega_r^-$. Let \begin{gather*} U_1 =\big\{u\in H_0^1(\Omega)\backslash\{0\};\frac{1}{\|u\|}t^- \big(\frac{u}{\|u\|}\big)>1\big\}\cup\{0\} ;\\ U_1 =\big\{u\in H_0^1(\Omega)\backslash\{0\};\frac{1}{\|u\|}t^- \big(\frac{u}{\|u\|}\big)<1\big\} . \end{gather*} Then $S_\lambda^-$ disconnects $H_0^1(\Omega)$ into two connected components $U_1$ and $U_2$. Moreover, $H_0^1(\Omega)\backslash S_\lambda^-=U_1\cup U_2$. For each $u\in S_\lambda^+$, we have \[ 10$ such that \[ 0\|u_\lambda^+\|^2+|c^2-\|u_\lambda^+\|^2|+2t_\lambda\int_\Omega u_\lambda^+ \upsilon_\varepsilon(x-y)dx \\ &>c^2>\big[t^-\big(\frac{u_\lambda^++t_\lambda\upsilon_\varepsilon(x-y)} {\|u_\lambda^++t_\lambda\upsilon_\varepsilon(x-y)\|}\big)\big]^2, \end{align*} that is $u_\lambda^++t_\lambda\upsilon_\varepsilon(x-y)\in U_2$. Thus there exists $0