\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2005(2005), No. 13, pp. 1--13.\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 2005 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2005/13\hfil Positive solutions to quasilinear equations] {Positive solutions to quasilinear equations involving critical exponent on perturbed annular domains} \author[C. O. Alves\hfil EJDE-2005/13\hfilneg] {Claudianor O. Alves} \address{Claudianor O. Alves \hfill\break Universidade Federal de Campina Grande \\ Departamento de Matem\'atica e Estat\'{\i}stica \\ CEP: 58109-970 Campina Grande - PB, Brazil} \email{coalves@dme.ufcg.edu.br} \date{} \thanks{Submitted August 5, 2004. Published January 30, 2005.} \thanks{Supported by Instituto do Mil\^enio and PADCT} \subjclass[2000]{35B33, 35H30} \keywords{p-Laplacian operator; critical exponents; deformation lemma} \begin{abstract} In this paper we study the existence of positive solutions for the problem $$-\Delta_{p}u=u^{p^{*}-1} \quad \hbox{in } \Omega \quad \hbox{and} \quad u=0 \quad \hbox{on } \partial{\Omega}$$ where $\Omega$ is a perturbed annular domain (see definition in the introduction) and $N>p \geq 2$. To prove our main results, we use the Concentration-Compactness Principle and variational techniques. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \section{Introduction} Consider the problem $$\label{Pl} \begin{gathered} -\Delta _pu= \lambda |u|^{p-2}u+| u| ^{p^{*}-2}u, \quad\mbox{in } \Omega\\ u > 0, \quad \mbox{in } \Omega \\ u=0, \quad \mbox{on } \partial{\Omega} \end{gathered}$$ where $\Omega \subset \mathbb{R}^{N}$ is a bounded domain with smooth boundary, $\lambda \geq 0$, $p^{*}=\frac{Np}{N-p}$, $N>p \geq 2$ and $$\Delta_{p}{u}= \sum_{j=1}^{N}\frac{\partial}{\partial x_{j}} \Big( |\nabla u |^{p-2} \frac{\partial u}{\partial x_{j}}\Big).$$ We recall that the weak solutions of \eqref{Pl} are critical points, on $W^{1,p}_0(\Omega)$, of the energy functional $$I_{\lambda}(u)=\frac{1}{p}\int_{\Omega} ( |\nabla{u}|^{p}-\lambda (u_{+})^{p} )dx -\frac{1}{p^{*}}\int_{\Omega}(u_{+})^{p^{*}}dx\,,$$ where $u_{+}(x)=\max\{u(x),0 \}$. Using Sobolev embedding it follows that $I_{\lambda} \in C^{1}(W^{1,p}_0(\Omega),\mathbb{R})$. An important point related with problem \eqref{Pl} it is Pohozaev's identity (see \cite{GV} and \cite{P}), which implies that \eqref{Pl} does not have a solution if $\Omega$ is strictly star-shaped with respect to the origin in $\mathbb{R}^{N}$ and $\lambda \leq 0$. Since the embedding $W_0^{1}(\Omega) \hookrightarrow L^{p^{*}} (\Omega)$ is not compact, we encounter serious difficulties in applying standard variational techniques to problem \eqref{Pl}. The lack of compactness can be understood by the fact that $I_{\lambda}$ does not satisfy the so-called Palais-Smale (PS) condition in the whole $\mathbb{R}$. Brezis and Nirenberg in \cite{BN} studied \eqref{Pl} for the case $p=2$ and $\lambda >0$,they used the fact that the (PS) condition holds in some energy range, for example in the interval $(-\infty, \frac{1}{N}S^{N/2})$, where $S$ is the best constant of the embedding $D^{1,p}(\mathbb{R}^{N})\hookrightarrow L^{p^{*}}(\mathbb{R}^{N})$ given by $$S=\min_{u\in D^{1,p}(\mathbb{R}^N) ,\; u \neq 0} \frac{\int_{\mathbb{R}^N}| \nabla u| ^pdx} {\big(\int_{\mathbb{R}^N}| u| ^{p^{*}}dx\big) ^{p/p^*}}.$$ Using the family of functions $\Phi _{\delta ,y}(x)=\frac{\big[ N\big( \frac{ N-p}{p-1}\big) \delta \big] ^{\frac{N-p}{p^2}}}{\big[ \delta +| x-y| ^{\frac p{p-1}}\big] ^{\frac{N-p}p}},\quad x,y\in \mathbb{R}^N,\; \delta >0$ which satisfies $\| \Phi _{\delta ,y}(x)\|_{1,p}^p=| \Phi _{\delta ,y}(x)|_{p^{*}}^{p^{*}}=S^{N/p}$ (see Talenti \cite{T}), where $\|u\|_{1,p}=\Big( \int_{\mathbb{R}^{N}}| \nabla u |^{p}dx \Big)^{1/p} \quad \mbox{and} \quad |u|_{p^{*}}= \Big(\int_{\mathbb{R}^{N}} |u|^{p^{*}}dx \Big)^{1/p^*}\, ,$ the authors in \cite{BN} showed that the minimization problem $S_{\lambda}= \min_{u \in W_0^{1,p}(\Omega) } \frac { \int_{\Omega} (| \nabla u |^{p}dx - \lambda |u|^{p} )dx }{ ( \int_{\Omega}|u|^{p^{*}}dx )^{\frac{p}{p^{*}}}}$ has a solution, hence \eqref{Pl} has a solution. After the results obtained in \cite{BN}, several authors have considered \eqref{Pl}, for instance, Struwe in \cite{S} (see also \cite{S1}) studied the behaviour of the Palais-Smale sequence of $I_{\lambda}$ for the case $p=2$ showing a result of Global Compactness. In his arguments, he used strongly some estimates for the Laplacian operator proved by Lions and Magenes in \cite{LM}. In \cite{C}, Coron used the study made in \cite{S} and proved that $(P)_{0}$ has a solution for a class of annular-shaped domains. In the papers of Bahri and Coron \cite{BhC}, Benci and Cerami \cite{BC} and Willem \cite{Wi} some results of existence of solution depending of the topology of $\Omega$ were proved.For the case $p \geq 2$, Gueda and Veron \cite{GV} and Garcia Azorero and Peral Alonso \cite{GA1} showed that the results obtained in \cite{BN} are true for p-Laplacian operator. There exists a rich literature involving the problem \eqref{Pl} with $p \geq 2$,we refer the reader to Peral Alonso \cite{I} and references therein. The main purpose of the present paper is to show that the result proved by Struwe in \cite{S} holds for the p-Laplacian operator and as a consequence the result obtained by Coron in \cite{C} is also true for the p-Laplacian operator with $p \geq 2$. To state our main result we need some definitions and notation. An important problem in this paper is the limit problem in $\mathbb{R}^{N}$ given by $$\label{Pi} \begin{gathered} - \Delta_{p}w=w^{p^{*}-1} \quad \mbox{in } \mathbb{R}^{N} \\ w >0 \quad \mbox{in } \mathbb{R}^{N} \\ w \in D^{1,p}(\mathbb{R}^{N}). \end{gathered}$$ Hereafter, let us denote by $I_{\infty}:D^{1,p}(\mathbb{R}^{N}) \to \mathbb{R}$ the energy functional related to limit problem, that is $$I_{\infty}(u)=\frac{1}{p}\int_{\mathbb{R}^{N}} |\nabla{u}|^{p}dx-\frac{1}{p^{*}}\int_{\mathbb{R}^{N}}(u_{+})^{p^{*}}dx.$$ We say that a domain $\Omega$ is a {\it Perturbed Annular Domain} (PAD) if there exist $R_{1},R_{2}>0$ such that $\Omega \supset \{ x \in \mathbb{R}^{N} : R_{1}< |x| 0 is a ball with center at y \in \mathbb{R}^{N} and radius s and B_{s}=B_{s}(0). For each a \in \mathbb{R}^{N}  we define the following sets:  \{ x^{N}>a \}= \{ x=(x^{1},\dots ,x^{N}) \in \mathbb{R}^{N}; x^{N}>a \}, \{x^{N}=a \}=\{ x=(x^{1},\dots ,x^{N}) \in \mathbb{R}^{N}; x^{N}=a \}  and  \{x^{N}a \} ) be a nonnegative solution of the problem  \begin{gathered} - \Delta_{p}v=v^{p^{*}-1} \quad \mbox{in } \{ x^{N}>a \} \\ v \geq 0 \quad \mbox{in } \{x^{N}>a \} \\ v=0 \quad \mbox{on } \{ x^{N}=a \}. \end{gathered}  Then v=0. \end{proposition} \begin{proof} By results showed by Trudinger \cite{Tr}, Guedda and Veron \cite{GV}, DiBenedetto \cite{DB}, and Tolksdorf \cite{To}, we have \[ v \in D_{0}^{1,2}(\{x^{N} > 0\}) \cap C^{1}(\{x^{N} \geq 0\})$ and adapting the ideas explored by Li and Shusen \cite{LS} $v(x) \to 0 \,\,\, \mbox{as} \,\,\, |x| \to \infty.$ Moreover, with suitable modifications, the arguments used by Esteban and Lions \cite{EL} and Gueda and Veron \cite[Theorem 1.1]{GV} show that $\int_{x^{N}=0}\langle x-x_0,\eta \rangle |v_{\eta}|^{p}d \sigma=0\,$ where $x_0$ is a point fixed in $\{x^{N}>0\}$ and $\eta$ is the forward normal to $\{x^{N}=0\}$. Hence $v_{\eta}=0$ on $\{x^{N}=0\}$ and by a result showed in V\asquez \cite{V} we have $v \equiv 0$. \end{proof} \begin{remark} \label{rmk1} \rm Proposition \ref{prop1} holds for sets of the form $\{ x^{N}0$ by well known arguments we get that $\nu_{i} \geq S^{N/p}$. From the definition of the function $v_{n}$ $\frac{1}{2}S^{N/p}= \sup_{y \in \mathbb{R}^{N} } \int_{B_{1}(y)}(v_{n})^{p^{*}}dx \geq \int_{B_{1}(y_{i})}(v_{n})^{p^{*}}dx$ then passing to the limit in the above inequality and using again Lemma \ref{lem1}, we obtain a contradiction. Thus, $\Lambda$ is empty and $\int_{\mathbb{R}^{N}}(v_{n})^{p^{*}}\Phi dx \to \int_{\mathbb{R}^{N}}(v_0)^{p^{*}}\Phi dx \ \ \forall \Phi \in C_0^{\infty}(\mathbb{R}^{N}) \ \mbox{as} \ n \to \infty$ which implies $v_{n} \to v_0$ in $L_{\rm loc}^{p^{*}}(\mathbb{R}^{N})$, consequently $\int_{B_{1}}(v_0)^{p^{*}}dx=\frac{1}{2}S^{N/p}$ and $v_0 \neq 0$. Using the fact the $v_0$ is not zero we have that $\lambda_{n} \to 0$, because if there exists $\delta >0$ such that $\lambda_{n} \geq \delta$, we have the following inequality $\int_{\mathbb{R}^{N}}(v_{n})^{p}dx= \frac{1}{\lambda_{n}^{p}} \int_{\mathbb{R}^{N}}(u_{n})^{p}dx \leq C_{1} \int_{\Omega }(u_{n})^{p}dx$ and by the fact that $u_{n} \to 0$ in $L^{p}(\Omega)$ it follows that $\int_{\mathbb{R}^{N}}(v_0)^{p}dx=0,$ which is a contradiction. Now, using the fact that $\lambda_{n} \to 0$ we may assume that there exists $x_0 \in \overline{\Omega}$ such that $x_{n} \to x_0 \in \overline{\Omega}$. By weak continuity of $v_{n}$ and (\ref{eq5}), the function $v_0$ is a solution of the problem $$\begin{gathered} -\Delta_{p}v=v^{p^{*}-1}, \quad \mbox{in } \Omega_{\infty} \\ v \geq 0, v \not\equiv 0 \quad \mbox{in } \Omega_{\infty} \\ v=0, \quad \mbox{on } \partial{\Omega_{\infty}}. \end{gathered}$$ To determine $\Omega_{\infty}$, we have to consider two cases: \begin{itemize} \item[(A)] $\frac{1}{\lambda_{n}}\mathop{\rm dist}(x_{n}, \partial \Omega ) \to \infty$ as $n \to \infty$ \item[(B)] $\frac{1}{\lambda_{n}}dist(x_{n}, \partial \Omega ) \leq \alpha$ for all $n \in \mathbb{N}$ and some $\alpha> 0$. \end{itemize} \noindent\textbf{Claim:} Case (B) above does not hold. In fact, assume by contradiction that (B) holds and that without loss of generality $x_{n} \to 0 \in \partial{\Omega}$. Moreover, we will suppose also that $0, \Omega$ and $\partial{\Omega}$ are described in the following form (see more details in Adimurthi, Pacella and Yadava \cite{AFY}): \begin{quote} There exist $\delta >0,$ an open neighborhood $\mathcal{N}$ of 0 and a diffeomorphism $\Psi:B_{\delta}(0) \to \mathcal{N}$ which has a jacobian determinant at 0 equal to one, with $\Psi (B_{\delta}^{{+}})=\mathcal{N} \cap \Omega$ where $B_{\delta}^{+}=B_{\delta}(0) \cap \{x^{N}>0 \}$. \end{quote} Now, let us define the function $\xi_{n} \in D^{1,p}(\mathbb{R}^{N})$ given by $$\xi_{n}(x)=\begin{cases} \lambda_{n}^{\frac{N-p}{p}}u_{n}(\Psi(\lambda_{n}x+P_{n})) \chi(\Psi(\lambda_{n}x+P_{n})),& x\in B_{\frac{\delta}{\lambda_{n}}} (-\frac{P_{n}}{\lambda_{n}}) \\ 0, & x\in \mathbb{R}^{N} \setminus B_{\frac{\delta}{\lambda_{n}}}(-{\frac{P_{n}}{\lambda_{n}}}) \end{cases}$$ where $\Psi(P_{n})=x_{n}$, $\chi \in C_{0}^{\infty}(\mathbb{R}^{N})$, $0 \leq \chi(x) \leq 1$ for all $x \in \mathbb{R}^{N}$, $\chi(x)=1$ for all $x \in \mathcal{O}_{\frac{\delta}{2}}$, $\chi(x)=0$ for all $x \in \mathcal{O}_{\frac{3 \delta}{4}}$, $\mathcal{O}_{\frac{\delta}{2}}=\Psi(B_{\frac{\delta}{2}})$, and $\mathcal{O}_{\frac{3\delta}{4}}=\Psi(B_{\frac{3\delta}{4}})$. By a simple computation, it is possible to show that for some subsequence $\frac{P^{N}_{n}}{\lambda_{n}} \to \alpha_{0} \quad \mbox{for some } \alpha_{0} \geq 0 \mbox{ as } n \to \infty$ and that there exists a nonnegative function $\xi \in D_0^{1,p}(\{x^{N}>-\alpha_{0} \})$ such that $\xi_{n}(x) \rightharpoonup \xi$ in $D^{1,p}(\mathbb{R}^{N})$ which satisfies $$\label{Pa0} \begin{gathered} - \Delta_{p}\xi=\xi^{p^{*}-1} \quad \mbox{in } \{x^{N} > -\alpha_{0}\} \\ \xi=0 \quad \mbox{on } \{x^{N}=- \alpha_{0} \}. \end{gathered}$$ From Proposition \ref{prop1}, we have that $\xi \equiv 0$. On the other hand, $$\int_{B_{1}}v_{n}^{p}dx \leq C \int_\mathcal{A}\xi_{n}^{p}dx$$ for all large $n$ where $\mathcal{A} \subset \{x^{N}> - \alpha_{0} \}$ is a bounded domain. Since $\{ \xi_{n} \}$ is a bounded sequence in $W^{1,p}( \mathcal{A })$, we obtain by Sobolev embedding $\int_{\mathcal{A} }\xi_{n}^{p}dx \to 0$ thus $\int_{B_{1}}v_{n}^{p}dx \to 0,$ and so $v_0 \equiv 0$ in $B_{1}$, which is a contradiction. Thus Case (A) holds, so $\Omega_{\infty}=\mathbb{R}^{N}$ and $v_0$ is a solution of \eqref{Pi}. To conclude, we consider $\Phi \in C_{0}^{\infty}(\mathbb{R}^{N})$ verifying $0 \leq \Phi(x)\leq 1$, $\Phi \equiv 1$ in $B_{1}$ and $\Phi =0$ in $B_{2}^{c}$. Let $w_{n}=u_{n}(x)-\lambda_{n}^{\frac{p-N}{p}}v_0(\frac{1}{\lambda_{n}}(x-x_{n})) \Phi(\frac{1}{\overline{\lambda_{n}}}(x-x_{n}))$ where we choose $\overline{\lambda_{n}}$ verifying $\widetilde{\lambda}_{n}=\frac{\lambda_{n}}{\overline{\lambda_{n}}} \to 0$. Considering $\widetilde{w}_{n}(x)=\lambda_{n}^{\frac{N-p}{p}}w_{n}(\lambda_{n}x+x_{n}) =v_{n}(x)-v_0(x)\Phi(\widetilde{\lambda}_{n}{x})$ and by repeating of the same arguments explored by Struwe in \cite{S}, we complete the proof of Lemma \ref{lem3}. \end{proof} \section{Proof of Theorem \ref{thm1}} By hypothesis we have $u_{n}(x)\to u_0(x)$ a.e in $\Omega$. Thus using standard arguments found in \cite{GA1,GV,I,A1}, we have $I'_{\lambda}(u_0)=0$. Suppose that $u_n$ does not converge to $u_0$ in $W^{1,p}_0(\Omega)$ and let $\{z_{n,1}\}\subset W^{1,p}_0(\Omega)$ be given by $z_{n,1}=u_n-u_0$. Then $z_{n,1} \rightharpoonup 0 \quad \mbox{but}\quad z_{n,1} \not\to 0 \mbox{ in } W_0^{1,p}(\Omega)\,.$ By Brezis and Lieb \cite{BL} and by Lemma \ref{lem2} it follows that \begin{gather} I_{0}(z_{n,1})=I_{\lambda}(u_n)-I_{\lambda}(u_0)+o_n(1)\,, \label{eq6}\\ I_{0}'(z_{n,1})=I_{\lambda}'(u_n)-I_{\lambda}' (u_0)+o_n(1)\,. \label{eq7} \end{gather} From these two equations, we conclude that $\{z_{n,1}\}$ is a $(PS)_c$ sequence for $I_{0}$. By Lemma \ref{lem3}, there exist $(\lambda_{n,1}) \subset \mathbb{R}$, $(x_{n,1}) \subset\mathbb{R}^{N}$, $z_1 \in D^{1,p}(\mathbb{R}^N)$ a non-trivial solution of \eqref{Pi} and a $(PS)_c$ sequence $\{z_{n,2} \}$ in $W_0^{1,p}(\Omega)$ for $I_{0}$ given by $z_{n,2}(x)=z_{n,1}(x)-\lambda_{n,1}^{\frac{p-N}{p}}z_1(\frac{1}{\lambda_{n,1}} (x-x_{n,1}))+o_n(1).$ If we define $v_{n,1}(x)=\lambda_{n,1}^{\frac{p-N}{p}}z_{n,1}(\lambda_{n,1}x+x_{n,1})$ and $\{\widetilde{z}_{n,2} \}$ by $\widetilde{z}_{n,2}(x)=v_{n,1}(x)-z_{1}(x)+o_n(1),$ we conclude by arguments explored in the proof of Lemma \ref{lem3} that $v_{n,1}\rightharpoonup z_1$ in $D^{1,p}(\mathbb{R}^{N})$, $I_{\infty}(v_{n,1})=I_{0}(z_{n,1})$ and $\|I_{0}'(v_{n,1},\Omega_{n,1})\|=o_n(1),$ where $\Omega_{n,1}=\frac{1}{\lambda_{n,1}}(\Omega - x_{n,1})$. Using again \cite{BL} and Lemma \ref{lem2}, we conclude that $I_{\infty}(\widetilde{z}_{n,2})=I_{\infty}(v_{n,1})-I_{\infty} (z_1)+o_n(1)=I_{\lambda}(u_n)-I_{\lambda}(u_0)-I_{\infty}(z_1)+o_n(1)$ and $\|I_{0}'(\widetilde{z}_{n,2}, \Omega_{n,1})\| \leq \|I_{0}'(v_{n,1}, \Omega_{n,1}) \| + \| I_{\infty}'(z_1) \|+o_n(1),$ consequently $\|I_{0}'(\widetilde{z}_{n,2},\Omega_{n,1})\|=o_{n}(1)$ and $\|I_{0}'(z_{n,2})\|=o_{n}(1)$. If $z_{n,2} \to 0$ in $W_0^{1,p}(\Omega)$ the theorem finishes. Now, if $\{ z_{n,2} \}$ does not converge to $0$ in $W_0^{1,p}(\Omega)$, we apply again Lemma \ref{lem3} and find $(\lambda_{n,2}) \subset\mathbb{R},\;(x_{n,2}) \subset \mathbb{R}^{N},z_2\in D^{1,p}(\mathbb{R}^N)\;$a non-trivial solution of \eqref{Pi} and a $(PS)_c$ sequence $\{z_{n,3} \}\;$ in $W_0^{1,p}(\Omega)$ for $I_{0}$ given by $z_{n,3}(x)=\widetilde{z}_{n,2}(x) -\lambda_{n,2}^{\frac{p-N}{p}}z_2(\frac{1}{\lambda_{n,2}}(x-x_{n,2}))+o_n(1).$ Considering the sequences $\{ v_{n,2} \}$ and $\{\widetilde{z}_{n,3} \}$ given by $v_{n,2}(x)=\lambda_{n,2}^{\frac{p-N}{p}}\widetilde{z}_{n,2} (\lambda_{n,2}x+x_{n,2})\quad \mbox{and}\quad \widetilde{z}_{n,3}(x)=v_{n,2}(x)-z_2(x)+o_n(1)$ we have $v_{n,2} \rightharpoonup z_2$ in $D^{1,p}(\mathbb{R}^{N})$ and $I_{\infty}(\widetilde{z}_{n,3})=I_{\infty}(\widetilde{z}_{n,2})-I_{\infty} (z_2)+o_n(1)=I_{\lambda}(u_n)-I_{\lambda}(u_0)-I_{\infty} (z_1)-I_{\infty}(z_2)+o_n(1)$ and $\| I_{0}'(\widetilde{z}_{n,3}, \Omega_{n,2})\| \leq \|I_{0}'(v_{n,2}, \Omega_{n,2}) \| + \|I_{\infty}'(z_2)\|+o_n(1),$ whence $\|I_{0}'(\widetilde{z}_{n,3},\Omega_{n,2})\|=o_{n}(1)$ and $\|I_{0}'(z_{n,3})\|=o_{n}(1)$. If $z_{n,3} \to 0$ the proof is done, if not, we repeat the arguments used, and then we will find $z_1,\dots ,z_k$ non-trivial solutions to \eqref{Pi} satisfying \begin{gather} \| \widetilde{z}_{n,k} \| ^p=\| u_n\| ^p-\| u_0\| ^p-\sum_{j=1}^{k-1}\| z_j\|_{1,p} ^p+o_n(1)\,, \label{eq8} \\ I_{\infty}(\widetilde{z}_{n,k})=I_{\lambda}(u_n)-I_{\lambda}(u_0) -\sum_{j=1}^{k-1}I_{\infty}(z_j)+o_n(1) \,. \label{eq9} \end{gather} Now, we recall that $$\| z_j \|_{1,p} ^p\geq S^{\frac Np}\;\;j=1,\dots ,k\,. \label{eq10}$$ Combining (\ref{eq8}) and (\ref{eq10}), $$0\leq \| \widetilde{z}_{n,k} \| ^p\leq \| u_n\| ^p-\| u_0\| ^p-\sum_{j=1}^{k-1}S^{\frac Np}=\| u_n\| ^p-\| u_0\| ^p-(k-1)S^{\frac Np}+o_n(1). \label{eq11}$$ Since $\{u_n\}$ is bounded, from \eqref{eq11} there exists $k\in \mathbb{N}$ such that $\limsup_{n\to \infty }\| \widetilde{z}_{n,k} \|^p\leq 0$. Consequently, $\widetilde{z}_{n,k} \to 0$ in $W_0^{1,p}(\Omega)$ and this concludes the proof. %\end{proof} \begin{corollary} \label{coro1} Let $\{u_n\}$ be a $(PS)_c$ sequence for $I_{\lambda}$ with $c\in (0,\frac 1NS^{\frac Np})$. Then $\{u_n\}$ contains a subsequence strongly convergent in $W_0^{1,p}(\Omega)$. \end{corollary} \begin{corollary} \label{coro2} The functional $I_{\lambda}:W_0^{1,p}(\Omega) \to \mathbb{R}$ satisfies the $(PS)_{c}$ condition in the interval $(\frac{1}{N}S^{N/p},\frac{2}{N}S^{N/p})$. \end{corollary} \begin{corollary} \label{coro3} Let $\{u_n\}$ be a $(PS)_c$ sequence for $I_{\lambda}$ with $c\in (\frac kNS^{\frac Np},\frac{(k+1)}NS^{N/p})$ and $k\in \mathbb{N}$. Then the weak limit $u_0$ of $\{u_n\}$ is not zero. \end{corollary} Hereafter we denote by $f_{\lambda}:W_0^{1,p}(\Omega)\to \mathbb{R}$ the functional $f_{\lambda}(u)=\int_{\Omega}(| \nabla u|^p- \lambda (u_{+})^p)dx$ and by $\mathcal{M}\subset W_0^{1,p}(\Omega)$ the manifold $\mathcal{M}=\{u\in W_0^{1,p}(\Omega);\int_{\Omega} (u_{+})^{p^{*}}dx=1\}.$ We remark that if $\{u_n\}\subset \mathcal{M}$ satisfies $f_{\lambda}(u_n)\to c \quad\mbox{and}\quad f_{\lambda}' | _\mathcal{M} (u_n)\to 0$ it follows that $\{v_n\}=\{c^{\frac{N-p}{p^2}}u_n\}\subset W_0^{1,p}(\Omega)$ satisfies the limits $I_{\lambda}(v_n)\to \frac 1Nc^{\frac Np}\quad\mbox{and}\quad I'_{\lambda}(v_n) \to 0 .$ \begin{corollary} \label{coro4} If there exist $\{u_n\}\subset \mathcal{M}$ and $c \in (S,2^{\frac{p}{N}}S)$ such that $f_{\lambda}(u_n)\to c$ and $f_{\lambda}'|_\mathcal{M}(u_n)\to 0$, then $f_{\lambda}$ has a critical point $u \in \mathcal{M}$ with $f_{\lambda}(u)=c$. \end{corollary} \begin{remark} \label{rmk2} \rm Corollary \ref{coro4} implies that \eqref{Pl} has at least a positive solution. \end{remark} \section{Proof of Theorem \ref{thm2}} Postponing the proof of Theorem \ref{thm2} for a moment, we first fix some notations and show some technical lemmas. In this section, we assume that $R_{1}=(4R)^{-1}<1<4R=R_{2}$ and denote by $\Sigma$ the unit sphere on $\mathbb{R}^{N}$, $\Sigma = \{ x \in \mathbb{R}^{N} : |x|=1 \}\,.$ For each $\sigma \in \Sigma$ and $t \in [0,1)$, we define the function $u_{t}^{\sigma} \in D^{1,p}(\mathbb{R}^{N})$ by $u_{t}^{\sigma}(x)= \Big[ \frac{1-t}{(1-t)^{\frac{p}{p-1}}+|x-t \sigma |^{\frac{p}{p-1}}} \Big]^{\frac{N-p}{p}}.$ Using the well known result obtained in \cite{T}, it follows that $S$ is attained on any such function $u_{t}^{\sigma}$.Moreover, letting $t \to 0$ we have $u_{t}^{\sigma} \to u_{0}= \Big[ \frac{1}{1+|x|^{\frac{p}{p-1}}} \Big]^{\frac{N-p}{p}} \quad \mbox{in } D^{1,p}(\mathbb{R}^{N})$ for any $\sigma \in \Sigma$. In the sequel $\phi \in C_0^{\infty}(\Omega)$ is a radially symmetric function such that $0 \leq \phi \leq 1$ on $\Omega, \phi \equiv 1$ on the annulus $\{ x \in \mathbb{R}^{N}: \frac{1}{2}< |x|<2 \}$ and $\phi \equiv 0$ outside the annulus $\{ x \in \mathbb{R}^{N}: \frac{1}{4}<|x|<4 \}$. Let us consider for $R \geq 1$ the functions $\phi_{R}(x)= \begin{cases} \phi(Rx) , & 0 \leq |x| < R^{-1} \\ 1, & R^{-1} \leq |x| < R \\ \phi(\frac{x}{R}), & |x| \geq R \,.\\ \end{cases}$ and $w_{t}^{\sigma}=u_{t}^{\sigma} \phi_{R}$, $w_0=u_0 \phi_{R} \in W_0^{1,p}(\Omega)$. \begin{lemma} \label{lem4} For each $\epsilon >0$ there exists $R>0$ such that $\int_{B_{R^{-1}}}(u_{t}^{\sigma})^{p^{*}}dx,\int_{B_{R^{-1}}}|\nabla u_{t}^{\sigma}|^{p}dx,\int_{B^{c}_{R}}(u_{t}^{\sigma})^{p^{*}}dx , \int_{B^{c}_{R}}|\nabla u_{t}^{\sigma}|^{p}dx < \epsilon$ uniformly in $\sigma \in \Sigma$ and $t \in [0,1)$. \end{lemma} \begin{proof} Using the definition of $u_{t}^{\sigma}$, we obtain $\int_{B_{R^{-1}}}(u_{t}^{\sigma})^{p^{*}}dx = (1-t)^{N} \int_{B_{R^{-1}}} \frac{dx}{ \big[ (1-t)^{\frac{p}{p-1}}+|x-t \sigma |^{\frac{p}{p-1}} \big]^{N} }$ or equivalently $\int_{B_{R^{-1}}}(u_{t}^{\sigma})^{p^{*}}dx =(1-t)^{N} \int_{B_{R^{-1}}(-t \sigma )} \frac{dy}{ \big[ (1-t)^{\frac{p}{p-1}}+|y|^{\frac{p}{p-1}} \big]^{N} }.$ Thus given $\epsilon >0$, there exists $\delta >0$ such that for all $t \in [1-\delta,1]$ and for all $R \geq R_0$, we have $$\int_{B_{R^{-1}}}(u_{t}^{\sigma})^{p^{*}}dx \leq (1-t)^{N}\int_{B_{R^{-1}}(-t \sigma)}\frac{dy}{|y|^{\frac{Np}{p-1}}}< \frac{\epsilon}{2} \ \forall \sigma \in \Sigma . \label{eq12}$$ On the other hand, there exists $R_0>0$ such that for all $R\geq R_0$, $$\int_{B_{1/(1-t)R}(\frac{-t \sigma }{1-t})}\frac{dw}{\big[1+|w|^{\frac{p}{p-1}} \big]^{N}}< \frac{\epsilon}{2} \ \ \forall \sigma \in \Sigma \quad \mbox{and} \quad \forall t \in [0,1-\delta] \label{eq13}$$ Hence, if $R_0$ is sufficiently large, from (\ref{eq12}) and (\ref{eq13}) $\int_{B_{R^{-1}}}(u_{t}^{\sigma})^{p^{*}}dx < \epsilon \ \forall t \in [0,1) \quad \mbox{and} \quad \forall \sigma \in \Sigma \mbox{ if } R \geq R_0\,.$ Now, we estimate the integral $\int_{{B_{R}}^{c}}{(u_{t}^{\sigma})}^{p^{*}}dx$: Note that $\int_{B^{c}_{R}}(u_{t}^{\sigma})^{p^{*}}dx=(1-t)^{\frac{N(p-2)}{p-1}} \int_{\Theta^{c}_{t}}\frac{dy}{\big[ 1+|y|^{\frac{p}{p-1}} \big]^{N} }\,,$ where $\Theta_{t}={B_{\frac{R}{(1-t)}}(\frac{-t\sigma}{{1-t}})}$; thus $\int_{{B^{c}_{R}}}(u_{t}^{\sigma})^{p^{*}}dx \leq C\int_{{B^{c}_{R-1}}}\frac{dy}{\big[ 1+|y|^{\frac{p}{p-1}} \big]^{N}}$ then for $R$ large, $\int_{B^{c}_{R}}(u_{t}^{\sigma})^{p^{*}}dx \leq \epsilon \quad \forall \sigma \in \Sigma ,\; \forall t \in [0,1).$ The estimates for the two integrals involving gradient of $u_{t}^{\sigma}$ follow with the same type of argument. \end{proof} As a consequence of the above lemma, we get the following result \begin{lemma} \label{lem5} The functions $\{ w_{t}^{\sigma }\}$ are strongly convergent in $D^{1,p}( \mathbb{R}^{N} )$ to $\{u_{t}^{\sigma} \}$ as $R \to \infty$ uniformly in $\sigma \in \Sigma$ and $t \in [0,1]$. Moreover, for each $R>0$ fixed, we have that $\{ w_{t}^{\sigma } \}$ is strongly convergent in $D^{1,p}( \mathbb{R}^{N} )$ to $\{ u_{t}^{\sigma} \}$ as $t \to 1$, uniformly in $\sigma \in \Sigma$. \end{lemma} \begin{remark} \label{rmk3} \rm Lemma \ref{lem5} also holds for the normalized functions $v_{t}^{\sigma}=w_{t}^{\sigma}/|w_{t}^{\sigma}|_{p^*}$; that is, $\| v_{t}^{\sigma}- \frac{u_{t}^{\sigma}}{|u_{t}^{\sigma}|_{p^{*}}} \|_{1,p} \to 0$ as $R \to \infty$ uniformly in $\sigma \in \Sigma$ and $t \in [0,1)$. \end{remark} Hereafter we define the function $\beta : \mathcal{M} \to \mathbb{R}^{N}$ namely `Barycenter'', by setting $\beta(u)= \int_{\Omega}x(u_{+})^{p^{*}}dx.$ \begin{proposition} \label{prop2} If $(u_{n}) \subset \mathcal{M}$ is such that $\|u_{n}\|^{p} \to S$, then $\mathop{\rm dist}(\beta(u_{n}),\overline{\Omega}) \to 0$. \end{proposition} \begin{proof} Note that the sequence $w_{n}=S^{\frac{N-p}{p^{2}}}u_{n}$ satisfies $I_{0}(w_{n}) \to \frac{1}{N}S^{N/p} \quad \mbox{and} \quad I'_{0}(w_{n}) \to 0\,.$ Using the fact that $S$ is never attained in a bounded domain, we get by Theorem \ref{thm1} that $w_{n} \rightharpoonup 0$ in $W_0^{1,p}(\Omega)$ and that there exists $(\lambda_{n}) \subset \mathbb{R}$, $(x_{n}) \subset \mathbb{R}^{N}$ with $x_{n} \to x_0 \in \overline{\Omega}$ and $v_0 \in \mathcal{M}$ such that $u_{n}(x)=\lambda_{n}^{\frac{p-N}{p}}v_0(\frac{1}{\lambda_{n}} (x-x_{n}))+o_{n}(1)\,.$ Then $\beta(u_{n})=\int_{\Omega}\frac{x}{\lambda_{n}^{N}}v_0(\frac{1}{\lambda_{n}} (x-x_{n}))^{p^{*}}dx + o_{n}(1).$ If $\phi \in C_0^{\infty}(\mathbb{R}^{N},\mathbb{R}^{N})$ is a function with $\phi (x)=x$ for $x \in \overline{\Omega}$, we get $\beta(u_{n})=\int_{\mathbb{R}^{N}}\phi (\lambda_{n}x+x_{n})v_0^{p^{*}}dx + o_{n}(1)\,.$ Then by Lebesgue's Theorem, $\int_{\mathbb{R}^{N}}\phi (\lambda_{n}x+x_{n})v_0^{p^{*}}dx\to \int_{\mathbb{R}^{N}}\phi (x_0)v_0^{p^{*}}dx=x_0 \in \overline{\Omega}$ whence $\mathop{\rm dist}(\beta(u_{n}),\overline{\Omega}) \to 0$. \end{proof} \begin{proof}[Proof of Theorem \ref{thm2}] Observe that by Lemma \ref{lem5} $f_{0}(v_{t}^{\sigma}) \to S$ as $R \to \infty$ uniformly in $\sigma \in \Sigma$ and $t \in [0,1)$. In particular, if $R\geq 1$ is sufficiently large, we have $\sup_{\sigma \in \Sigma ,\; t \in [0,1)} f_{0}(v_{t}^{\sigma})0. Thus, f_0 does not have a critical value in the interval (S, 2^{\frac{p}{N}}S). Moreover, by Theorem \ref{thm1}, f_0 verifies on \mathcal{M} the (PS)_{c} condition in (S, 2^{\frac{p}{N}}S). Using the same arguments explored in \cite{S1}, there exist \delta >0 and a flow  \Phi : \mathcal{M} \times [0,1] \to \mathcal{M} such that \[ \Phi ( \mathcal{M}_{S_{1}},1 ) \subset \mathcal{M}_{S+\delta }$ where $\mathcal{M}_{c}=\{ u \in \mathcal{M}:f_{0}(u) \leq c \},\quad \Phi(u,t)=u \quad \forall u \in \mathcal{M}_{S + \frac{\delta}{2}}.$ Using Proposition \ref{prop2}, we can assume that $\beta ( \mathcal{M}_{S+ \delta} ) \subset U$, where $U$ is a neighborhood of $\overline{\Omega}$ such that any point $p \in U$ has a unique nearest neighbor $q=\pi(p) \in \Omega$ and such that the projection $\pi$ is continuous . The map $h: \Sigma \times [0,1] \to \Omega$ given by $$h(\sigma ,t)= \begin{cases} \pi ( \beta ( \Phi ( v_{t}^{\sigma},1))), & t \in [0,1) \\ \sigma , & t=1 \end{cases}$$ is well-defined. Furthermore, $h$ is a continuous function in $\Sigma \times [0,1]$, which is obvious for $t \in [0,1)$, now for the case $t=1$ we use the following argument: Note that for each $(\sigma_{n},t_{n}) \in \Sigma \times [0,1]$, we compute \begin{align*} \int_{\Omega}x(u_{t_{n}}^{\sigma_{n}})^{p^{*}}dx & = (1-t_{n})^{\frac{N(p-2)}{p-1}}t_{n}\sigma_{n}\int_{\Omega_{t_{n}}} \frac{dx}{ \big[ 1 + |w|^{\frac{p}{p-1}} \big]^{N} }\\ &\quad +(1-t_{n})^{\frac{N(p-2)}{p-1}}(1-t_{n})\int_{\Omega_{t_{n}}}\frac{wdx}{ {\big[ 1 + |w|^{\frac{p}{p-1}} \big]}^{N} }\,, \end{align*} where $\Omega_{t_{n}}=\frac{(\Omega-t_{n}\sigma_{n})}{1-t_{n}}$. Since ${|u_{t_{n}}^{\sigma_{n}}|}_{p^{*}}^{p^{*}}=(1-t_{n})^{\frac{N(p-2)} {p-1}}\int_{\mathbb{R}^{N}} \frac{dx}{\big[ 1 + |w|^{\frac{p}{p-1}} \big]^{N} },$ if $(\sigma_{n},t_{n}) \to (\sigma,1)$ as $n \to \infty$ we get $\beta \Big( \frac{u_{t_{n}}^{\sigma_{n}}}{|u_{t_{n}}^{\sigma_{n}}|_{p^{*}}} \Big) \to \sigma .$ Using the limit above together with Lemma \ref{lem5}, we conclude that $\lim_{n \to \infty}h(\sigma_{n},t_{n})= \sigma = h(\sigma,1)\,.$ Therefore, $h$ is a continuous functions in $\Sigma \times [0,1]$. Also observe that \begin{gather*} h(\sigma,0)= \pi ( \beta ( \Phi (v_{0},1)))=x_{0} \in \Omega, \quad \forall \sigma \in \Sigma \\ h(\sigma ,1)= \sigma, \quad \forall \sigma \in \Sigma \end{gather*} hence $h$ is a contraction of $\Sigma$ in $\Omega$, contradicting the hypotheses on $\Omega$. \end{proof} \subsection*{Acknowledgment:} The author is grateful for the hospitality offered at IMECC - UNICAMP, where he was visiting while this work was done. Special thanks are given to professors Djairo G. de Figueiredo, and J. V. Goncalves for their suggestions about this manuscript. \begin{thebibliography}{00} \bibitem{AFY} Admurthi, F. Pacella, and S.L. Yadava; {\it Interaction between the geometry of the boundary and positive solutions of a semilinear Neumann problem with critical nonlinearity}, J. funct. Analysis 113(1993),318-350. \bibitem{A1} C. O. Alves; {\it Exist\^{e}ncia de solu\c {c}\~{a}o positiva de equa\c {c}\~{o}es el\'{\i}pticas n\~{a}o-lineares variacionais em $\mathbb{R}^N$,} Doct. Dissertation, UnB,1996. \bibitem{A2} C. O. Alves; {\it Existence of positive solutions for a problem with lack compactness involving the p-Laplacian,} Nonl. Analysis TMA, 51(2002), 1187-1206. \bibitem{BhC} A. Bahri and J.M. Coron; {\it On a Nonlinear elliptic equation involving the critical Sobolev exponent: The effect of the topology of the domain,} Comm. Pure Appl. Math. 41 (1988), 253-294. \bibitem{BC} V. Benci and G. Cerami; {\it The effect of the domain topology on the number of positive solutions of nonlinear elliptic problems,} Arch. Rat. Mech. Anal. 114 (1991),79-93. \bibitem{BN} H. Brezis and L. Nirenberg; {\it Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents,} Comm. Pure Appl. Math. 36(1983),486-490. \bibitem{BL} H. Brezis and E. Lieb; {\it A Relation between pointwise convergence of functions and convergence of functional,} Proc. Amer. Math. Soc. 88(1983),486-490. \bibitem{C} J. M. Coron; {\it Topologie et cas limite des injections de Sobolev.} C.R. Acad. Sc. Paris 299, Ser. I ( 1984) 209-212. \bibitem{DB} E. DiBenedetto; {\it $C^{1+\alpha}$ local regularity of weak solutions of degenerate elliptic equations,} Nonl. Analysis TMA, 7 (1983), 827-850. \bibitem{EL} M. J. Esteban and P. L. Lions; {\it Existence and non-existence results % for semilinear elliptic problems in unbounded domains,} Proceding of the Royal Society of Edinburgh, 93A ( 1982), 1-14. \bibitem{GA1} J. Garcia Azorero and I. Peral Alonso; {\it Multiplicity of solutions for elliptic problems with critical exponent on with a nonsymetric term,} Trans. Amer. Math. Society vol.323 2(1991), 877-895. \bibitem{GV} M. Gueda and L. Veron; {\it Quasilinear elliptic equations involving critical Sobolev exponents,} Nonlinear Anal. TMA 13(1989),419-431. \bibitem{L} P. L. Lions; {\it The concentration-compactness principle in the calculus of variations:The limit case,} Rev. Mat. Iberoamericana 1(1985), 145-201. \bibitem{LM} J. L. Lions and E. Magenes; {\it Non-homogeneous boundary value problems and applications} I. Grundlehren 181, Springer, Berlin-Heedelberg-New York (1972). \bibitem{LS} L. Gongbao and Y. Shusen; {\it Eigenvalue problems for quasilinear elliptic equations on $\mathbb{R}$,} Comm. Part. Diff. Equations, 14 (1989), 1291-1314. \bibitem{I} I. Peral Alonso; {\it Multiplicity of solutions for the p-Laplacian, Second School on Nonlinear Funtional Analysis and Applications to Differential Equations,} ICTP - Trieste ( Italy ) 1997. \bibitem{P} S. Pohozaev; {\it Eigenfunctions of the equation $\Delta{u}+\lambda f(u)=0$.} Soviet Math. Dokl. 6 (1965),1408-1411. \bibitem{S} M. Struwe; {\it A global compactness result for elliptic boundary value problem involving limiting nonlinearities}, Math. Z. 187(1984), 511-517. \bibitem{S1} M. Struwe; {\it Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems,} Springer Verlag (1990). \bibitem{T} G. Talenti; {\it Best constant in Sobolev inequality,} Ann Math. 110(1976), 353-372. \bibitem{V} J. L. V\'asquez; {\it A strong maximum principle for some quasilinear elliptic,} Appl. Math. Optim, 12(1984),191-202 \bibitem{Tr} N. S. Trudinger; {\it On Harnack type inequalities and their application to quasilinear elliptic equations,} Comm. Pura and Appl. Math., XX (1967), 721-747. \bibitem{To} P. Tolksdorf; {\it Regularity for a more general class of quasilinear elliptic equations,} J. Diff. Equations, 51 (1984), 126-150. \bibitem{Wi} W. Willem; {\it Minimax Theorems}, Birkhauser, 1986. \end{thebibliography} \end{document}