\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2007(2007), No. 84, pp. 1--6.\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 2007 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2007/84\hfil Elliptic problem with singular critical] {An existence result for elliptic problems with singular critical growth} \author[Y. Nasri\hfil EJDE-2007/84\hfilneg]{Yasmina Nasri} \address{Yasmina Nasri \newline Universit\'{e} de Tlemcen, d\'{e}partement de math\'{e}matiques, BP 119 Tlemcen 13000, Alg\'{e}rie} \email{y\_nasri@mail.univ-tlemcen.dz} \thanks{Submitted February 6, 2007. Published June 6, 2007.} \subjclass[2000]{35J20, 35J60} \keywords{Palais-Smale condition; singular potential; Sobolev exponent; \hfill\break\indent mountain-pass theorem} \begin{abstract} We prove the existence of nontrivial solutions for the singular critical problem $$-\Delta u-\mu \frac{u}{|x|^2}=\lambda f(x)u+u^{2^{\ast }-1}$$ with Dirichlet boundary conditions. Here the domain is a smooth bounded subset of $\mathbb{R}^N$, $N\geq 3$, and $2^{\ast }=\frac{2N}{N-2}$ which is the critical Sobolev exponent. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \section{Introduction} This paper concerns the semilinear elliptic problem $$\begin{gathered} -\Delta u-\mu \frac{u}{|x|^{2}}=\lambda f(x)u+u^{2^{\ast}-1} \quad \text{in }\Omega \\ u>0 \quad \text{in }\Omega \\ u=0 \quad \text{on }\partial \Omega , \end{gathered} \label{Plm}$$ where $\Omega$ is a smooth bounded domain in $\mathbb{R}^{N}$, $N\geq 3$ with $0\in \Omega$; $\lambda$ and $\mu$ are positive parameters with $0\leq \mu <\overline{\mu }:=(\frac{N-2}{2})^{2}$, $\overline{\mu }$ is the best constant in the Hardy inequality, $2^{\ast }=\frac{2N}{N-2}$ is the critical Sobolev exponent and $f$ is a positive measurable function which will be specified later. In recent years, many people have paid much attention to the existence of nontrivial solutions for singular problems we cite \cite{CR,GP,J,T} and the references cited therein. For $f( x)=1$, Jannelli \cite{J} obtained the following results: If $0\leq \mu \leq \overline{\mu }-1$, then \eqref{Plm} has at least one solution $u\in H_{0}^{1}( \Omega )$ for all $0<\lambda <\lambda _{1}( \mu )$ where $\lambda _{1}( \mu)$ is the first eigenvalue of the operator $( -\triangle -\frac{\mu }{|x|^{2}})$ in $H_{0}^{1}( \Omega)$. If $\overline{\mu }-1<\mu <\overline{\mu }$, then \eqref{Plm} has at least one solution $u\in H_{0}^{1}( \Omega )$ for all $\mu ^{\ast }<\lambda <\lambda _{1}( \mu )$ where \begin{equation*} \mu ^{\ast }=\min_{\varphi \in H_{0}^{1}( \Omega )} \frac{\int_{\Omega } \frac{|\nabla \varphi ( x)| ^{2}}{|x|^{2\sigma }}dx}{\int_{\Omega } \frac{|\varphi ( x)|^{2}}{|x|^{2\sigma }}dx} \end{equation*} and $\sigma =\sqrt{\overline{\mu }}+\sqrt{\overline{\mu }-\mu }$. If $\overline{\mu }-1<\mu <\overline{\mu }$ and $\Omega =B( 0,R)$ then \eqref{Plm} has no solution for $\lambda \leq \mu ^{\ast }$. If $\lambda \leq 0$ and $\Omega$ is star shaped then \eqref{Plm} has no nontrivial solutions using Pohozaev-type identity. For the quasi-linear form of \eqref{Plm} the problem has been studied by \cite{GP} for $\mu =0$ and $f( x)=\frac{1}{|x|^{q}}$ where $0\leq q0$ such that $J_{\lambda ,\mu }( u) \geq \alpha$ for all $u\in H_{0}^{1}(\Omega)$ such that $\left\Vert u\right\Vert _{\mu }=\delta$ for all $0<\lambda <\lambda _{\mu }^{1}( f)$. $2/J_{\lambda ,\mu }(v) <0$ for all $v\in H_{0}^{1}( \Omega )$ such that $\Vert v\Vert _{\mu }>\delta$. \end{lemma} \begin{proof} Using the definition of $S_{\mu }$ and the fact that $0<\lambda <\lambda _{\mu }^{1}(f)$, we obtain \begin{equation*} J_{\lambda ,\mu }(u)\geq \frac{1}{2}\big(1-\frac{\lambda }{\lambda _{\mu }^{1}(f)}\big)\Vert u\Vert _{\mu }^{2}-\frac{1}{2^{\ast }(S_{\mu })^{2^{\ast }/2}}\Vert u\Vert _{\mu }^{2^{\ast }}. \end{equation*} So for $\delta >0$ sufficiently small there exists $\alpha >0$ such that \begin{equation*} J_{\lambda ,\mu }(u)\geq \alpha \quad \text{for }\Vert u\Vert _{\mu }=\delta . \end{equation*} For $t>0$, \begin{equation*} J_{\lambda ,\mu }(tu)=\frac{t^{2}}{2}(\Vert u\Vert _{\mu }^{2}-\int_{\Omega }f(x)u^{2}dx)-\frac{t^{2^{\ast }}}{2^{\ast }}\Vert u\Vert _{2^{\ast }}^{2^{\ast }}dx, \end{equation*} as $t\to +\infty$ we have $J_{\lambda ,\mu }(tu)\to -\infty$. Then there exists $v\in H_{0}^{1}(\Omega)$ such that $J_{\lambda ,\mu }(v)<0$ for $\Vert v\Vert _{\mu }>\delta$. \end{proof} \begin{lemma} \label{lem6} Assume that $0<\lambda <\lambda _{\mu }^{1}(f)$ and $0\leq \mu \leq \overline{\mu }-(\frac{2-\beta }{2})^{2}$. Then $\sup_{0\leq t<\infty } J_{\lambda ,\mu }( tv_{\varepsilon })<\frac{1}{N}( S_{\mu })^{N/2}$ provided $\varepsilon >0$ is a small enough. \end{lemma} \begin{proof} Consider the functions \begin{equation*} g\left( t\right) :=J_{\lambda ,\mu }(tv_{\varepsilon })=\frac{t^{2}}{2}% (\Vert v_{\varepsilon }\Vert _{\mu }^{2}-\lambda \int_{\Omega }f(x)v_{\varepsilon }^{2}dx)-\frac{t^{2^{\ast }}}{2^{\ast }}, \end{equation*} where $v_{\varepsilon }$ is the extremal function defined in \eqref{a1}. Note that $\lim_{t\to +\infty }g(t)=-\infty$ and $g(t)>0$ when $t$ is close to $0$. So that $\sup_{t\geq 0}g(t)$ is attained for some $t_{\varepsilon }>0$. From \begin{equation*} 0=g'(t_{\varepsilon })=t_{\varepsilon }\big(\Vert v_{\varepsilon }\Vert _{\mu }^{2}-\lambda \int_{\Omega }f(x)v_{\varepsilon }^{2}dx\big) -t_{\varepsilon }^{2^{\ast -1}}\Vert v_{\varepsilon }\Vert _{2^{\ast }}^{2^{\ast }}, \end{equation*} we have \begin{equation*} \;t_{\varepsilon }=\Big[\Vert v_{\varepsilon }\Vert _{\mu }^{2}-\lambda \int_{\Omega }f(x)v_{\varepsilon }^{2}dx\Big]^{\frac{1}{2^{\ast }-2}}. \end{equation*} Thus, \begin{equation*} g\left( t_{\varepsilon }\right) =\frac{1}{N}\Big(\Vert v_{\varepsilon }\Vert _{\mu }^{2}-\lambda \int_{\Omega }f(x)v_{\varepsilon }^{2}dx\Big) ^{\frac{2^{\ast }}{2^{\ast }-2}}. \end{equation*} Then as in \cite{J} (see also \cite{Ch}), we have the following estimates: \begin{equation*} \int_{\Omega }\Big( |\nabla v_{\varepsilon }|^{2}dx-\mu \frac{v_{\varepsilon }{}^{2}}{|x|^{2}}\Big) dx =S_{\mu }^{\frac{N}{2}}+C\varepsilon ^{\frac{N-2}{2}}; \end{equation*} since $f\in \mathcal{F}_{2,\beta }$, there exist $r>0$ and $C_{1},C_{2}>0$ such that $K_{1}|x|^{-\beta }\leq f(x) \leq K_{2}|x|^{-\beta }$ on $B_{R}\left( 0\right)$. Thus \begin{gather*} C_{1}\varepsilon ^{\frac{\sqrt{\bar{\mu}}}{2\sqrt{\bar{\mu}-\mu }}(2-\beta )}\leq \int_{\Omega }f(x)v_{\varepsilon }^{2}dx\leq C_{2}\varepsilon ^{\frac{\sqrt{\bar{\mu}}}{2\sqrt{\bar{\mu}-\mu }}(2-\beta )} \quad \text{if }\mu <\overline{\mu }-(\frac{2-\beta }{2})^{2}; \\ C_{1}\varepsilon ^{\frac{N-2}{2}}|\log \varepsilon |\leq \int_{\Omega }f(x)v_{\varepsilon }^{2}dx\leq C_{2}\varepsilon ^{\frac{N-2}{2}}|\log \varepsilon | \quad \text{if }\mu =\overline{\mu }-(\frac{2-\beta }{2})^{2}. \end{gather*} Consequently, \begin{equation*} g\left( t_{\varepsilon }\right) \leq \begin{cases} \frac{1}{N}S_{\mu }^{\frac{N}{2}}+C\varepsilon ^{\frac{N-2}{2}} -C_{1}\varepsilon ^{\frac{N-2}{2}}|\log \varepsilon | & \text{if }\mu =\overline{\mu }-(\frac{2-\beta }{2})^{2}, \\ \frac{1}{N}S_{\mu }^{\frac{N}{2}}+C\varepsilon ^{\frac{N-2}{2}} -C_{1}\varepsilon ^{\frac{\sqrt{\bar{\mu}}}{2\sqrt{\bar{\mu}-\mu }}(2-\beta )} & \text{if }\mu <\overline{\mu }-(\frac{2-\beta }{2})^{2}. \end{cases} \end{equation*} Therefore, for $\varepsilon >0$ sufficiently small and $\mu \leq \overline{\mu }-(\frac{2-\beta }{2})^{2}$ we get \begin{equation*} \sup_{t\geq 0}J_{\lambda ,\mu }(tv_{\varepsilon }) <\frac{1}{N}S_{\mu }^{N/2}. \end{equation*} \end{proof} \begin{proof}[Proof of Theorem \protect\ref{thm1}] From Lemmas \ref{lem4}, \ref{lem5} and \ref{lem6}, $J_{\lambda ,\mu }$ satisfies all assumptions of mountain pass Theorem \cite{AR}, then $c$ is a critical value i.e. there exists $u\in H_{0}^{1}(\Omega)$ such that $J_{\lambda ,\mu }'(u)=0$ and $J_{\lambda ,\mu }(u)=c>0$. 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