\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 214, pp. 1--12.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2013 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2013/214\hfil Ground state solutions] {Ground state solutions for semilinear problems with a Sobolev-Hardy term} \author[X. Chen, W. Chen \hfil EJDE-2013/214\hfilneg] {Xiaoli Chen, Weiyang Chen} % in alphabetical order \address{Xiaoli Chen \newline Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China} \email{littleli\_chen@163.com} \address{Weiyang Chen (corresponding author)\newline Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China} \email{xiaowei19901207@126.com} \thanks{Submitted April 19, 2013. Published September 26, 2013.} \subjclass[2000]{35J60, 35J65} \keywords{Existence; ground state; critical Hardy-Sobolev exponent; \hfill\break\indent semilinear Dirichlet problem} \begin{abstract} In this article, we study the existence of solutions to the problem \begin{gather*} -\Delta u= \lambda u+\frac{|u|^{2_s^\ast-2}u}{|y|^s}, \quad x\in \Omega,\\ u = 0, \quad x\in \partial \Omega, \end{gather*} where $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$ $(N\geq3)$. We show that there is a ground state solution provided that $ N=4$ and $\lambda_m<\lambda<\lambda_{m+1}$, or that $N\geq5$ and $\lambda_m\leq\lambda<\lambda_{m+1}$, where $\lambda_m$ is the m'th eigenvalue of $-\Delta$ with Dirichlet boundary conditions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \allowdisplaybreaks \section{Introduction} Let $\Omega$ be a smooth bounded domain of $\mathbb{R}^N=\mathbb{R}^k\times\mathbb{R}^{N-k}$, where $2\leq k0 \quad \text{in } \mathbb{R}^N,\quad u\in D^{1,2}(\mathbb{R}^N) \end{equation} up to a constant. If $s=0$, Equation \eqref{eq:1.2} becomes the Sobolev inequality, for which best constant was computed, and proved existence of minimizers in \cite{Aubin} and \cite{Talenti}. In the case $s=2$, \eqref{eq:1.2} still holds true, it is an extension of the Hardy inequality. In the more general case $0\leq s<2$ with $k=N$, the best constant was obtained in \cite{GGMT}, and minimizers were found in \cite{Lieb}, which are radially symmetric. Therefore, it can be shown by using ODEs, see \cite{Lieb}, that up to dilations and translations, minimizers take the form \[ \frac 1{(1+|x|^{2-s})^{\frac {N-2}{2-s}}}. \] It is noted that equation \eqref{eq:1.3} is invariant with respect to the scalings and $z$-translations; that is, $u$ is a solution of \eqref{eq:1.3} if only if $u_\alpha(x) = \alpha^{(N-2)/2}u(\alpha y,\alpha(z-z_0))$, $\alpha>0$, satisfies the equation. Hence, problem \eqref{eq:1.3} has lack of the compactness. In the case $0< s <2$, $2\leq k0$, and $S$ is achieved by the concentration-compactness principle. So problem \eqref{eq:1.3} has a positive solution in $D^{1,2}(\mathbb{R}^N)$. Since the minimizer of problem \eqref{eq:1.2} can not be radially symmetric, they cannot be found among solutions of ODEs, but of PDEs. This brings difficulties to find exact forms of the minimizer. In the particular case $s=1$, problem \eqref{eq:1.3} becomes \begin{equation}\label{eq:1.4} -\Delta u=\frac{u^{\frac{N}{N-2}}}{|y|},\quad u>0 \quad \text{in } \mathbb{R}^N,\quad u\in D^{1,2}(\mathbb{R}^N). \end{equation} By the moving plane method, it was proved in \cite{FMS} that all solutions of \eqref{eq:1.4} are cylindrically symmetric. Thus, problem \eqref{eq:1.4} can be reduced to an elliptic equation in the positive cone in $\mathbb{R}^2$, and it was shown in \cite{FMS} that $u$ is a solution of \eqref{eq:1.4} if and only if \begin{equation}\label{eq:1.5} u(y,z)=\lambda^{(N-2)/2}V(\lambda y,\lambda(z+z_0)) \end{equation} for some $\lambda>0$ and $z_0\in\mathbb{R}^{N-k}$, where \begin{equation}\label{eq:1.6} V(x)=V(y,z)=\frac{C_{N,k}}{\left((1+|y|)^2+|z|^2\right)^{(N-2)/2}} =\frac{\left((N-2)(k-1)\right)^{(N-2)/2}}{\left((1+|y|)^2 +|z|^2\right)^{(N-2)/2}}. \end{equation} This result allows one to obtain existence results for problem \eqref{eq:1.1} in the case $s = 1$. Denote by $0<\lambda_1,\dots,\lambda_k,\dots$ the eigenvalues of $-\Delta$ with zero Dirichlet boundary condition. When $0<\lambda<\lambda_1$ and $s = 1$, it was proved in \cite{AWZ} and \cite{FabbriPHD} that there exists a solution of problem \eqref{eq:1.1} by the mountain pass lemma and constrained variation respectively. In this article, we consider the existence of solutions to problem \eqref{eq:1.1} for general case $0C_1>0$ such that \begin{equation}\label{eq:1.7} \frac{C_1}{1+|x|^{N-2}}\leq u(x)\leq\frac{C_2}{1+|x|^{N-2}}. \end{equation} This estimate suffices to serve our purpose. Using the Nehari manifold method introduced in \cite{P}, and developed in \cite{SW}, we show the following result. \begin{theorem}\label{thm:1.0} Let $ N=4$ and $\lambda_m<\lambda<\lambda_{m+1}$ or $N\geq5$ and $\lambda_m\leq\lambda<\lambda_{m+1}$ for some $m\in\mathbb{N}$, then there exists a ground state solution of problem \eqref{eq:1.1}. \end{theorem} In section 2, we describe a variational framework to study the ground state solution of problem \eqref{eq:1.1}. We prove Theorem \ref{thm:1.0} in section 3. \section {Preliminaries} Denote by $E=H^1_0(\Omega)$ the Hilbert space with the scalar product \[ \langle u,v\rangle=\int_\Omega\nabla u\nabla v\,dx \] and the induced norm $\|\cdot\|$. Let $(\varphi_j,\lambda_j)$ be the eigenfunctions and eigenvalues of $-\Delta$ in $\Omega$ with zero Dirichlet boundary condition. Suppose that $m$ is a fixed positive integer and $\lambda_m\leq\lambda<\lambda_{m+1}$, we define the subspaces $E^-=\operatorname{span}\{\varphi_1,\dots,\varphi_m\}$ and $E^+=\operatorname{span}\{\varphi_j,j\geq m\}$ of $E$, then $E=E^+\oplus E^-$. The functional associated to problem \eqref{eq:1.1} is defined by \[ J(u)=\frac12\int_\Omega|\nabla u|^2\,dx -\frac\lambda2\int_\Omega|u|^2\,dx -\frac1{2_s^\ast}\int_\Omega\frac{|u|^{2_s^\ast}}{|y|^s}\,dx \] for $u\in H^1_0(\Omega)$, which is $C^1$ and critical points of $J$ are solutions of problem \eqref{eq:1.1}. To find ground state solutions of \eqref{eq:1.1}, we introduce as \cite{P} a submanifold of $E$. Define \begin{equation}\label{eq:2.1} \mathcal{N}=\{u\in E\setminus{\{0\}}:\langle {\nabla J(u)},u\rangle=0, {\nabla J(u)}\in E^+ \}. \end{equation} The set $\mathcal{N}$ is the intersection of the standard Nehari manifold $\{u\in E\setminus{\{0\}}:\langle {\nabla J(u)},u\rangle=0\}$ with the pre-image $(\nabla J)^{-1}(E^+)$. \begin{proposition}\label{prop:2.1} The set $\mathcal{N}$ is a $C^1$ submanifold of $E$ with codimension $m+1$. Moreover, every critical point of the restriction $J|_\mathcal{N}$ is a nontrivial critical point of the functional $J$. \end{proposition} \begin{proof} The result can be proved as \cite{SWW}, see also \cite{S}. We sketch the proof here for reader's convenience. Let $F:E\setminus{\{0\}}\to \mathbb{R}\times E^-$ be a map defined by \[ F(u)=(\langle {\nabla J(u)},u\rangle,Q\nabla J(u)), \] where $Q$ is the orthogonal projection of $E$ onto $E^-$, then $\mathcal{N}=F^{-1}(0)$. Consider the inner product \[ (t_1,z_1)\cdot(t_2,z_2)=t_1t_2+\langle z_1,z_2\rangle \quad \text{for } t_1,t_2\in \mathbb{R},\; z_1,z_2\in E^-. \] We claim that for every $(t,z)\in \mathbb{R}\times E^-$, $(t,z)\neq(0,0)$, the inequality \begin{equation}\label{eq:2.2} (DF(u)(tu+z))\cdot(t,z)<0 \end{equation} holds. This implies the first part of the proposition. Now, we prove the claim. Indeed, for $(t,z)\neq(0,0)$, since $$ \langle {\nabla J(u)},u\rangle=\langle {\nabla J(u)},z\rangle=0, $$ we deduce that \begin{equation}\label{eq:2.3} \begin{split} &(DF(u)(tu+z))\cdot(t,z)\\ &=\Big(\int_{\Omega}|\nabla z|^2\, dx-{\lambda}\int_{\Omega}|z|^2 \,dx\Big)\\ &\quad-\int_{\Omega}\Big((2_s^\ast-2)t^2|u|^2+2(2_s^\ast-2)tzu +(2_s^\ast-1)|z|^2\Big)\frac{|u|^{2_s^\ast-2}}{|y|^s}dx. \end{split} \end{equation} For $\lambda_m\leq\lambda<\lambda_{m+1}$, it is readily verified that \eqref{eq:2.2} holds. Next, we verify as in \cite{SWW} that $w\in E$ is a critical point of $J$ if and only if $u\in \mathcal{N}$ and $DJ(u)|_{T_u\mathcal{N}}=0$. The proof is complete. \end{proof} We recall that a ground state solution $u$ to \eqref{eq:1.1} is any element of $\mathcal{N}$ such that $DJ(u)$ vanishes on $T_u\mathcal{N}$ and $J(u)=c$, where \begin{equation}\label{eq:2.4} c=\inf_{\mathcal{N}}J. \end{equation} By the argument in \cite{SW}, for every $v\in E^+\setminus{\{0\}}$, there is a unique continuous map pair $(f(v),g(v))\in (0,\infty)\times E^-$ such that $F(f(v)v+g(v))=0$ and \[ J(f(v)v+g(v))=\max_{t>0,z\in E^-}J(tv+z). \] Hence, \begin{equation}\label{eq:2.5} c=\inf_{\mathcal{N}}J=\inf_{v\neq0,v\in E^+}J(f(v)v+g(v)) =\inf_{\{v\neq0,v\in E^+\}}\max_{\{t>0,z\in E^-\}}J(tv+z). \end{equation} \section{Existence results} In this section, we show that problem \eqref{eq:2.4} is achieved. The minimizer of problem \eqref{eq:2.4} is actually a ground state solution of \eqref{eq:1.1}. Let \begin{equation}\label{eq:3.1} S=\inf_{u\in E,u\not=0}\bigg\{\frac{\int_{\Omega}|\nabla u|^2\,dx}{(\int_{\Omega}\frac{u^{2_s^\ast}}{|y|^s}\,dx)^{2/2_s^\ast}}\bigg\}. \end{equation} We know from \cite{BT} that $S$ can be achieved, which is independent of $\Omega$ and depends only by $N,k,s$, moreover the infimum $S$ is never achieved when $\Omega$ is a bounded domain, we denote the minimizer by $U(x)>0$. By \eqref{eq:1.7}, $$ \frac{C_1}{1+|x|^{N-2}}\leq U(x)\leq\frac{C_2}{1+|x|^{N-2}}. $$ The following elementary lemma is readily verified. \begin{lemma}\label{lem:3.1} Suppose $A>0$, $B>0$. Then \[ \max_{t>0}(A\frac{t^2}2-B\frac{t^{2_s^\ast}}{2_s^\ast}) =\frac{2-s}{2(N-s)}\Big(\frac{A}{B^{2/2_s^\ast}}\Big)^{\frac{N-s}{2-s}}. \] \end{lemma} \begin{lemma}\label{lem:3.2} Suppose that \begin{equation}\label{eq:3.2} c<\frac{2-s}{2(N-s)}S^{\frac{N-s}{2-s}}, \end{equation} then there exists $v\in E^+\setminus{\{0\}}$ such that \[ \max_{t>0,w\in E^-}J(tv+w)=J(f(v)v+g(v))=c. \] \end{lemma} \begin{proof} Take any sequence $\{v_n\}$ in $E^+\setminus{\{0\}}$ such that $\|v_n\|=1$ and \begin{equation}\label{eq:3.3} \max_{t>0,w\in E^-}J(tv_n+w)\to c. \end{equation} Without loss of generality, we can assume that \begin{gather*} v_n\rightharpoonup v \quad \text{in } E^+,\\ v_n \to v \quad \text{in } L^{2}(\Omega), \\ v_n\to v \quad\text{a.e. }\Omega. \end{gather*} Suppose \[ A=\lim_{n\to \infty}\int_{\Omega}|\nabla (v_n-v)|^2 dx\,,\quad B=\lim_{n\to \infty}\int_{\Omega}\frac{|v_n-v|^{2_s^\ast}}{|y|^s}dx\,. \] Using the Brezis-Lieb's Lemma, from \eqref{eq:3.3} we obtain \begin{equation}\label{eq:3.4} J(tv+w)+\frac 1 2 At^2-\frac 1{2_s^\ast}B t^{2_s^\ast}\leq c, \quad \forall t>0, \; \forall w\in E^-. \end{equation} If $v=0$ and $B=0$, from the assumption $\|v_n\|=1$, we deduce that $A=1$. Hence $t^2\leq2c-2J(w)$ for every $t>0$ and every $w\in E^-$, a contradiction. Assume now $B\neq0$. From Lemma \ref{lem:3.1}, we obtain that \begin{equation}\label{eq:3.5} \frac{2-s}{2(N-s)}S^{\frac{N-s}{2-s}}\leq\frac{2-s}{2(N-s)} \Big(\frac A {B^{ 2 /{2_s^\ast}}}\Big)^{\frac{N-s}{2-s}} =\max_{t>0}\Big(\frac 1 2 At^2-\frac 1{2_s^\ast}B t^{2_s^\ast}\Big). \end{equation} If $v=0$, we obtain from \eqref{eq:3.2}, \eqref{eq:3.4} and \eqref{eq:3.5} that \[ \frac{2-s}{2(N-s)}S^{\frac{N-s}{2-s}} \leq c<\frac{2-s}{2(N-s)}S^{\frac{N-s}{2-s}}, \] a contradiction. Thus $v\neq0$. Denote $h=g(v)/f(v)$. It follows from the definition of $c$ that \begin{equation}\label{eq:3.6} \begin{split} c&\leq J(f(v)(v+h))=\max_{t>0}J(t(v+h))\\ &=\frac{2-s}{2(N-s)}\bigg\{\frac {\int_{\Omega}|\nabla (v+h)|^2\,dx -{\lambda}\int_{\Omega}|v+h|^2dx}{ \big(\int_{\Omega} \frac{|v+h|^{2_s^\ast}}{|y|^s}dx\big)^{2/2_s^\ast}} \bigg\}^{\frac{N-s}{2-s}}. \end{split} \end{equation} By \eqref{eq:3.4} and Lemma \ref{lem:3.1}, \begin{equation}\label{eq:3.7} \begin{split} c&\geq \max_{t>0}\Big(J(t(v+h))+\frac 1 2 At^2-\frac 1{2_s^\ast}B t^{2_s^\ast} \Big)\\ &=\frac{2-s}{2(N-s)}\bigg\{\frac {A+\int_{\Omega}|\nabla (v+h)|^2\,dx -{\lambda}\int_{\Omega}|v+h|^2\,dx} { \big(B +\int_{\Omega}\frac{|v+h|^{2_s^\ast}}{|y|^s}\,dx\big)^{2/2_s^\ast}} \bigg\}^{\frac{N-s}{2-s}}. \end{split} \end{equation} Putting together \eqref{eq:3.2}, \eqref{eq:3.5}, \eqref{eq:3.6} and \eqref{eq:3.7}, we obtain \begin{equation}\label{eq:3.8} \begin{split} &\Big(\frac{2(N-s)}{2-s}c\Big)^{\frac{2-s}{N-s}} \Big(B +\int_{\Omega}\frac{|v+h|^{2_s^\ast}}{|y|^s}\,dx\Big)^{2/2_s^\ast}\\ &<\Big(\frac{2(N-s)}{2-s}c\Big)^{\frac{2-s}{N-s}} \Big(B^{2/2_s^\ast} +\Big(\int_{\Omega}\frac{|v+h|^{2_s^\ast}}{|y|^s}\,dx \Big)^{2/2_s^\ast}\Big)\\ &0$, we define $U_\varepsilon(x)=\varepsilon^{\frac{2-N}2}U(\frac{x-(0,z_0)}{\varepsilon})$, $u_\varepsilon=\varphi(x)U_\varepsilon(x)$, Then $u_\varepsilon\in E$ for $\varepsilon>0$ small. We have following estimates for $u_{\varepsilon}$. \begin{lemma}\label{lem:3.3} Suppose $N\geq3$, we have \begin{gather}\label{eq:3.9} \|u_{\varepsilon}\|^2=\|U\|^2+O(\varepsilon^{N-2})+O(\varepsilon^{N-s}),\\ \label{eq:3.10} \int_{\Omega}\frac{|u_{\varepsilon}|^{2_s^\ast}}{|y|^{s}}dx =\int_{R^N}\frac{|U|^{2_s^\ast}}{|y|^{s}}dx+O(\varepsilon^{N-s}),\\ \label{eq:3.11} \int_{\Omega}|u_{\varepsilon}(x)|^{2}dx \geq \begin{cases} C\varepsilon^{2}+O(\varepsilon^{N-2}),& N\geq5,\\ C\varepsilon^{2}|\ln\varepsilon|+O(\varepsilon^2),& N=4,\\ C\varepsilon+O(\varepsilon^2),& N=3, \end{cases} \\ \label{eq:3.12} \int_{\Omega}u_{\varepsilon}(x)dx\leq C\varepsilon^{(N-2)/2},\\ \label{eq:3.13} \int_{\Omega}\frac{|u_{\varepsilon}|^{2_s^\ast-1}}{|y|^{s}}dx \leq C\varepsilon^{(N-2)/2}. \end{gather} \end{lemma} \begin{proof} First, we estimate \eqref{eq:3.10}. There holds \begin{align*} \int_{\Omega}\frac{|u_{\varepsilon}|^{2_s^\ast}}{|y|^{s}}dx &=\int_\Omega\frac{|\varphi U_{\varepsilon}|^{2_s^\ast}}{|y|^{s}}dx =\int_{\Omega}\frac{U_{\varepsilon}^{2_s^\ast}}{|y|^{s}}dx-\int_{\Omega}(1-\varphi^{2_s^\ast})\frac{U_{\varepsilon}^{2_s^\ast}}{|y|^{s}}dx\\ &=\int_{R^{N}}\frac{U_{\varepsilon}^{2_s^\ast}}{|y|^{s}}dx-\int_{R^{N}\backslash\Omega}\frac{U_{\varepsilon}^{2_s^\ast}}{|y|^{s}}dx- \int_{\Omega}(1-\varphi^{2_s^\ast})\frac{U_{\varepsilon}^{2_s^\ast}}{|y|^{s}}dx\\ &=\int_{R^{N}}\frac{U^{2_s^\ast}}{|y|^{s}}dx-\int_{R^{N}\backslash\Omega}\frac{U_{\varepsilon}^{2_s^\ast}}{|y|^{s}}dx- \int_{\Omega\setminus B_{\frac\rho2}(0,z_0)}(1-\varphi^{2_s^\ast})\frac{U_{\varepsilon}^{2_s^\ast}}{|y|^{s}}dx.\\ \end{align*} Since \[ \int_{R^{N}\backslash B_{R}(0,z_{0})} \frac{U_{\varepsilon}^{2_s^\ast}}{|y|^{s}}dx \leq\int_{R^{N}\backslash\Omega}\frac{U_{\varepsilon}^{2_s^\ast}}{|y|^{s}}dx \leq\int_{R^{N}\backslash B_{\rho}(0,z_{0})} \frac{U_{\varepsilon}^{2_s^\ast}}{|y|^{s}}dx, \] while \begin{align*} \int_{R^{N}\backslash B_{R}(0,z_{0})}\frac{U_{\varepsilon}^{2_s^\ast}}{|y|^{s}}dx &=\int_{R^{N}\backslash B_{R}(0,z_{0})}\varepsilon^{s-N} \frac{U(\frac{x-(0,z_{0})}{\varepsilon})^{2_s^\ast}}{|y|^{s}}dx\\ &=\int_{R^{N}\backslash B_{R}(0)}\varepsilon^{s-N} \frac{U(\frac{x}{\varepsilon})^{2_s^\ast}}{|y|^{s}}dx\\ &\le C\varepsilon^{s-N}\int_{R^{N}\backslash B_{R}(0)} \Big(\frac{1}{1+|\frac x{\varepsilon}|^{N-2}}\Big)^{2_s^\ast}\frac1{|y|^{s}}dx\\ &= C\varepsilon^{N-s}\int_{R^{N}\backslash B_{R}(0)} \Big(\frac{1}{\varepsilon^{N-2}+|x|^{N-2}}\Big)^{2_s^\ast}\frac1{|y|^{s}}dx\\ &=O(\varepsilon^{N-s}), \end{align*} and similarly, \[ \int_{R^{N}\backslash B_{\rho}(0,z_{0})}\frac{U_{\varepsilon}^{2_s^\ast}}{|y|^{s}}dx=O(\varepsilon^{N-s}). \] Thus, we obtain \[ \int_{\Omega}\frac{|u_{\varepsilon}|^{2_s^\ast}}{|y|^{s}}dx=\int_{R^{N}}\frac{U^{2_s^\ast}}{|y|^{s}}dx+O(\varepsilon^{N-s}).\\ \] That is, \eqref{eq:3.10} holds. Next, we estimate \eqref{eq:3.11}. In fact, \begin{align*} \int_{\Omega}|u_{\varepsilon}|^{2}dx &=\int_{\Omega}\varphi^{2}|U_{\varepsilon}|^{2}dx \leq\int_{B_{\rho}(0,z_{0})}{|U_{\varepsilon}|}^{2}\,dx\\ &=\varepsilon^{2-N}\int_{B_{\rho}(0)}{U(\frac{x}{\varepsilon})}^{2}dx\\ &\leq\varepsilon^{2-N}\int_{B_{\rho}(0)} \frac{C}{{\left(1+|\frac{x}{\varepsilon}|^{N-2}\right)}^{2}}dx\\ &=\varepsilon^{N-2}\int_{B_{\rho}(0)}\frac{C}{{\left(\varepsilon^{N-2}+|x|^{N-2}\right)}^{2}}dx\\ &\leq\varepsilon^{N-2}\int_{B_{\varepsilon}(0)}\frac{C}{\varepsilon^{2(N-2)}}dx+\varepsilon^{N-2}\int_{B_{\rho}(0)\backslash B_{\varepsilon}(0)}\frac{C}{|x|^{2(N-2)}}dx\\ &=\begin{cases} C\varepsilon^{2}+O(\varepsilon^{N-2}),& N\geq5,\\ C\varepsilon^{2}|\ln\varepsilon|+O(\varepsilon^2),& N=4,\\ C\varepsilon+O(\varepsilon^2),& N=3. \end{cases} \end{align*} Now, we estimate \eqref{eq:3.9}. Observe that \[ \int_{\Omega}|\nabla u_{\varepsilon}|^{2}\,dx =\int_{\Omega}U_{\varepsilon}^{2}|\nabla\varphi|^{2}dx +\int_{\Omega}\nabla U_{\varepsilon}\nabla(\varphi^{2}U_{\varepsilon})\,dx \] and $-\Delta U_{\varepsilon}= U_{\varepsilon}^{2\ast-1}/|y|^{s}$, we find \[ \int_{\Omega}\nabla U_{\varepsilon}\nabla(\varphi^{2}U_{\varepsilon})\,dx =\int_{\Omega}\varphi^{2}\frac{U_{\varepsilon}^{2\ast}}{|y|^{s}}\,dx \] and \[ \int_{\Omega}|\nabla u_{\varepsilon}|^{2}\,dx =\int_{\Omega}|\nabla\varphi|^{2}U_{\varepsilon}^{2}\,dx +\int_{\Omega}\varphi^{2}\frac{U_{\varepsilon}^{2\ast}}{|y|^{s}}. \] Since $\nabla\varphi=0$ in $B_\rho(0,z_0)$, we have \begin{align*} \int_{\Omega}|\nabla\varphi|^{2}U_{\varepsilon}^{2}\,dx &=\int_{\Omega\setminus B_\rho(0,z_0)}|\nabla\varphi|^{2}U_{\varepsilon}^{2}\,dx\\ &\leq\int_{B_R(0,z_0)\setminus B_\rho(0,z_0)}|\nabla\varphi|^{2} U_{\varepsilon}^{2}\,dx\\ &\leq C\int_{B_R(0,z_0)\setminus B_\rho(0,z_0)}U_{\varepsilon}^{2}\,dx\\ &\leq C\int_{B_R(0)\setminus B_\rho(0)}\varepsilon ^{2-N} \frac 1{(1+|\frac{x}\varepsilon|^{N-2})^2}\,dx=O(\varepsilon^{N-2}). \end{align*} On the other hand, we can show that \[ \int_{\Omega}\varphi^{2}\frac{|U_{\varepsilon}|^{2_s^\ast}}{|y|^{s}}dx =\int_{R^{N}}\frac{U^{2_s^\ast}}{|y|^{s}}dx+O(\varepsilon^{N-s}) =\int_{R^N}|\nabla U|^2\,dx+O(\varepsilon^{N-s}). \] Therefore, \[ \int_{\Omega}|\nabla u_\varepsilon|^2\,dx =\|\nabla U\|^2+O(\varepsilon^{N-2})+O(\varepsilon^{N-s}). \] Now, we estimate \eqref{eq:3.12}. \begin{align*} \int_{\Omega}u_{\varepsilon}dx &=\int_{B_{\rho}(0,z_0)}\varepsilon^{\frac{2-N}{2}} U(\frac{x-(0,z_0)}{\varepsilon})\\ &\leq C\int_{B_{\rho}(0)}\varepsilon^{\frac{2-N}{2}}\frac{1}{1+| \frac{x}{\varepsilon}|^{N-2}}dx\\ &=\varepsilon^{\frac{N-2}{2}}\int_{B_{\rho}(0)} \frac{1}{\varepsilon^{N-2}+|x|^{N-2}}dx\\ &\leq C\varepsilon^{\frac{N-2}{2}}\int_{B_{\varepsilon}(0)} \frac{1}{\varepsilon^{N-2}}dx+ C\varepsilon^{\frac{N-2}{2}}\int_{B_{\rho}(0)\backslash{B_{\varepsilon}(0)}} \frac{1}{|x|^{N-2}}dx\\ &\leq C\varepsilon^{\frac{N-2}{2}}. \end{align*} Finally, there holds \begin{align*} \int_{\Omega}\frac{|u_{\varepsilon}|^{2_s^\ast-1}}{|y|^{s}}\,dx &\leq \int_{B_\rho(0,z_0)}\frac{|U_{\varepsilon}|^{2_s^\ast-1}}{|y|^{s}}\,dx\\ &\leq \varepsilon^{\frac{N+2-2s}2}\int_{B_\rho(0)} \Big(\frac{1}{\varepsilon^{N-2}+|x|^{N-2}}\Big)^{2_s^\ast-1}\frac{dx}{|y|^{s}}\\ &=\varepsilon^{\frac{N+2-2s}2}\int_{B_\varepsilon(0)} \Big(\frac{1}{\varepsilon^{N-2}+|x|^{N-2}}\Big)^{2_s^\ast-1} \frac{dx}{|y|^{s}}\\ &\quad+\varepsilon^{\frac{N+2-2s}2}\int_{{B_\rho(0)}\setminus{B_\varepsilon(0)}} \Big(\frac{1}{\varepsilon^{N-2}+|x|^{N-2}}\Big)^{2_s^\ast-1}\frac{dx}{|y|^{s}}\\ &\leq C\varepsilon^{(N-2)/2} +\varepsilon^{\frac{N+2-2s}2}\int_{{B_\rho(0)} \setminus{B_\varepsilon(0)}}\frac{1}{(|y|^2+|z|^2)^{\frac{N+2-2s}2}} \frac{dx}{|y|^{s}} \end{align*} and \begin{align*} &\varepsilon^{\frac{N+2-2s}2}\int_{{B_\rho(0)}\setminus{B_\varepsilon(0)}} \frac{1}{(|y|^2+|z|^2)^{\frac{N+2-2s}2}}\frac{dx}{|y|^{s}}\\ &=\varepsilon^{\frac{N+2-2s}2}\int_{({B_\rho(0)} \setminus{B_\varepsilon(0)})\cap\{x=(y,z):|y|\geq|z|\}} \frac{1}{(|y|^2+|z|^2)^{\frac{N+2-2s}2}}\frac{dx}{|y|^{s}}\\ &\quad+\varepsilon^{\frac{N+2-2s}2}\int_{({B_\rho(0)}\setminus{B_\varepsilon(0)})\cap\{x=(y,z):|y|<|z|\}}\frac{1}{(|y|^2+|z|^2)^{\frac{N+2-2s}2}}\frac{dx}{|y|^{s}}\\ &\leq C\varepsilon^{\frac{N+2-2s}2}\int_{\{x=(y,z):\frac{\varepsilon}{\sqrt 2}<|y|,|z|<\rho\}}\frac{1}{|z|^{N+2-2s}}\frac{dx}{|y|^{s}}\\ &\quad+C\varepsilon^{\frac{N+2-2s}2}\int_{\{x=(y,z):\frac{\varepsilon}{\sqrt 2}<|y|,|z|<\rho\}}\frac{1}{|y|^{N+2-2s}}\frac{dx}{|y|^{s}}\\ &\leq C\varepsilon^{(N-2)/2}, \end{align*} which implies \eqref{eq:3.13}. \end{proof} \begin{proposition}\label{prop:3.1} There holds \[ c<\frac{2-s}{2(N-s)}S^{\frac{N-s}{2-s}}. \] \end{proposition} \begin{proof} We will check that \begin{equation}\label{eq:3.14} \max_{t>0,v\in E^-}J(t u_\varepsilon+v)<\frac{2-s}{2(N-s)}S^{\frac{N-s}{2-s}}. \end{equation} Let $\omega=\Omega\setminus{\rm supp}\varphi$. By \cite[Lemma 3.3]{SWW}, $v\mapsto\|v\|_{L^{2_s^\ast}(\omega)}$ defines a norm on $E^-$. Since $\dim E^-=m<+\infty$, all the norms are equivalent on $E^-$. For every $t>0$ and every $v\in E^-$, by convexity we deduce \begin{equation}\label{eq:3.15} \begin{aligned} &\int_{\Omega}\frac{|tu_{\varepsilon}(x)+v(x)|^{2_s^\ast}}{|y|^s}\, dx\\ &=\int_{\Omega\setminus\omega}\frac{|tu_{\varepsilon}(x)+v(x)|^{2_s^\ast}}{|y|^s} \,dx+\int_{\omega}\frac{|v(x)|^{2_s^\ast}}{|y|^s}\, dx\\ &\geq t^{2_s^\ast}\int_{\Omega}\frac{|u_{\varepsilon}(x)|^{2_s^\ast}}{|y|^s}\, dx+{2_s^\ast}t^{2_s^\ast-1}\int_{\Omega}\frac{|u_{\varepsilon}(x)|^{2_s^\ast-1}v(x)}{|y|^s}\, dx +{2_s^\ast}C_1\|v\|^{2_s^\ast}. \end{aligned} \end{equation} It follows that \begin{equation}\label{eq:3.16} \begin{aligned} J(t u_\varepsilon+v) &\leq J(t u_\varepsilon)+t\int_{\Omega} \nabla u_\varepsilon\nabla v+{\frac 12}\int_{\Omega}|\nabla v|^2\,dx\\ &\quad -\lambda t\int_\Omega u_{\varepsilon}(x)v(x)\,dx -{\frac {\lambda} 2}\int_{\Omega}|v(x)|^2\,dx\\ &\quad-t^{2_s^\ast-1}\int_{\Omega}\frac{|u_{\varepsilon}(x)|^{2_s^\ast-1}v(x)}{|y|^s} dx-C_1\|v\|^{2_s^\ast}.\\ \end{aligned} \end{equation} By the assumption $\lambda_m\leq\lambda<\lambda_{m+1}$, \begin{equation}\label{eq:3.17} \int_{\Omega}|\nabla v|^2\,dx-{\lambda} \int_{\Omega}|v(x)|^2\,dx\leq(\lambda_m-\lambda)\|v\|^2\leq0. \end{equation} In particular, we can write \[ J(t u_\varepsilon+z)\leq A(t^2+t\|v\|+t^{2_s^\ast-1}\|v\|) -B(t^{2_s^\ast}+\|v\|^{2_s^\ast}) \] for suitable constants $A>0$ and $B>0$. Hence there exists $R>0$ such that, for $\varepsilon$ small, $t>R$ and $v\in E^-$ there holds $J(t u_\varepsilon+v)\leq0$. On the other hand, whenever $t\leq R$, \begin{equation}\label{eq:3.18} J(t u_\varepsilon+v)\leq J(t u_\varepsilon) +O\big(\varepsilon^{(N-2)/2}\big)\|v\| -C_1\|v\|^{2_s^\ast}\leq J(t u_\varepsilon) +O\Big(\varepsilon^{\frac {(N-2)(N-s)}{N+2-2s}}\Big). \end{equation} Indeed, integrating by parts and using the definition of $E^-$, we obtain \begin{equation}\label{eq:3.19} \begin{split} &\int_{\Omega}\nabla u_\varepsilon\nabla v\,dx -{\lambda}\int_{\Omega}u_{\varepsilon}(x) v(x)\,dx\\ &=\int_\Omega(-\Delta v)u_\varepsilon\,dx -{\lambda}\int_{\Omega}u_{\varepsilon}(x) v(x)\,dx\\ &\leq|\lambda_m-\lambda|\int_{\Omega}|u_\varepsilon(x)v(x)|\,dx \leq|\lambda_m-\lambda|\|v(\cdot)\|_{L^\infty}\int_{\Omega}|u_\varepsilon(x)|\,dx\\ &\leq C|\lambda_m-\lambda|\|v\|\int_{\Omega}|u_{\varepsilon}(x)|\,dx \end{split} \end{equation} and \begin{eqnarray*} \big|\int_{\Omega}\frac{|u_{\varepsilon}(x)|^{2_s^\ast-1}v}{|y|^s}\,dx\big| \leq C\|v\|\int_{\Omega}\frac{|u_{\varepsilon}(x)|^{2_s^\ast-1}}{|y|^s}\,dx. \end{eqnarray*} By \eqref{eq:3.12} and \eqref{eq:3.13} , we get \[ \int_{\Omega}|u_{\varepsilon}(x)|\,dx \leq C\varepsilon^{(N-2)/2}, \quad \int_{\Omega}\frac{|u_{\varepsilon}(x)|^{2_s^\ast-1}}{|y|^s}\,dx \leq C\varepsilon^{(N-2)/2} . \] By the Young inequality, \[ O\Big(\varepsilon^{(N-2)/2}\Big)\|v\| \leq O\Big(\varepsilon^{\frac {(N-2)(N-s)}{N+2-2s}}\Big)+C_1\|v\|^{2_s^\ast}. \] Therefore, together with \eqref{eq:3.17}, we see that \eqref{eq:3.18} holds. Since $N\geq 5$ implies $\frac {(N-2)(N-s)}{N+2-2s}>2$. By Lemma \ref{lem:3.2}, for $\varepsilon>0$ small enough, \begin{align*} &\max_{t>0, v\in E^-}J(t u_\varepsilon+v)\\ &\leq \max_{t>0}J(t u_\varepsilon) +O\Big(\varepsilon^{\frac {(N-2)(N-s)}{N+2-2s}}\Big)\\ &=\frac {2-s}{2(N-s)}\bigg(\frac {\| u_\varepsilon\|^2 -\lambda \|u_{\varepsilon}(x)\|^2_{L^2(\Omega)}} {(\int_{\Omega}\frac{|u_{\varepsilon}(x)|^{2_s^\ast}}{|y|^s}\,dx)^{2/2_s^\ast}} \bigg)^{\frac{N-s}{2-s}} +O\Big(\varepsilon^{\frac {(N-2)(N-s)}{N+2-2s}}\Big)\\ &\leq\frac {2-s}{2(N-s)}\Big(S-C\lambda\varepsilon^{2} +O(\varepsilon ^{N-2}) \Big)^{\frac{N-s}{2-s}} +O\Big(\varepsilon^{\frac {(N-2)(N-s)}{N+2-2s}}\Big)\\ &<\frac {2-s}{2(N-s)}S^{\frac{N-s}{2-s}}. \end{align*} Assume now that $N=4$. From \eqref{eq:3.12} and \eqref{eq:3.13} , we obtain \[ \int_{\Omega}|u_{\varepsilon}(x)|\,dx\leq C\varepsilon, \quad \int_{\Omega}\frac{|u_{\varepsilon}(x)|^{2_s^\ast-1}}{|y|^s}\,dx \leq C\varepsilon. \] By the assumption $\lambda_m<\lambda<\lambda_{m+1}$, \begin{equation}\label{eq:3.20} \int_{\Omega}|\nabla v|^2\,dx-{\lambda} \int_{\Omega}|v(x)|^2\,dx \leq(\lambda_m-\lambda)\|v\|^2=-C_2\|v\|^2. \end{equation} Inequality \eqref{eq:3.16}, \eqref{eq:3.19} and \eqref{eq:3.20} imply that, for $t\leq R$, \[ J(t u_\varepsilon+v)\leq J(t u_\varepsilon) +O(\varepsilon)\|v\|-C_2\|v\|^{2}\leq J(t u_\varepsilon)+O(\varepsilon^2). \] From Lemma \ref{lem:3.2}, for $\varepsilon>0$ small enough, we obtain \begin{align*} &\max_{t>0, v\in E^-}J(t u_\varepsilon+v)\\ &\leq\frac {2-s}{2(4-s)} \bigg(\frac {\|u_\varepsilon\|^2-\lambda \|u_{\varepsilon}\|^2_{L^2(\Omega)}} {(\int_{\Omega}\frac{|u_{\varepsilon}|^{2_s^\ast}}{|y|^s}\,dx)^{2/2_s^\ast}} \bigg)^{\frac{4-s}{2-s}} +O(\varepsilon^2)\\ &\leq\frac {2-s}{2(4-s)} \bigg(\frac{\|U\|^2+O\left(\varepsilon^2\right) -\lambda\left(C\varepsilon^2\left|\ln\varepsilon\right|+O(\varepsilon^2)\right)} {\big(\int_{R^N}\frac{|U|^{2_s^\ast}}{|y|^s}\,dx +O\left(\varepsilon ^{4-s}\right)\big)^{2/(4-s)} } \bigg)^{\frac {4-s}{2-s}} +O(\varepsilon^2)\\ &\leq\frac {2-s}{2(4-s)} \left(S-C\lambda\varepsilon^2\left|\ln\varepsilon\right|+O\left(\varepsilon ^2\right) \right)^{\frac{4-s}{2-s}}+O(\varepsilon^2)\\ &<\frac {2-s}{2(4-s)}S^{\frac{4-s}{2-s}}. \end{align*} \end{proof} \begin{proof}[Proof of Theorem \ref{thm:1.0}] By Lemma \ref{lem:3.1} and Proposition \ref{prop:3.1}, there exists $u\in \mathcal{N}$ such that $J(u)=c$ and $DJ(u)|_{T_u\mathcal{N}}=0$. It follows from Proposition \ref{prop:2.1} that $DJ(u)=0$ on $X$. \end{proof} \subsection*{Acknowledgments} Xiaoli Chen is supported by NNSF of China (grant 11261023), the foundation of teacher's division of Jiangxi Province (grant GJJ12203), Startup Foundation for Doctors of Jiangxi Normal University. WeiYang Chen is supported by NNSF of China (grant 11271170), GAN PO 555 program of Jiangxi, NSF of Jiangxi Province (grant 20122BAB201008) \begin{thebibliography}{99} \bibitem{AWZ} A. Al-aati, C. Wang, J. Zhao; \emph{Positive solutions to a semilinear elliptic equation with a Sobolev-Hardy term}, Nonlinear Analysis, \textbf{74} (2011), 4847--4861. \bibitem {Aubin} T. Aubin; \emph{probl\'{e}mes isop\'{e}rim\'{e}triques et espaces de Sobolev}, J. Differ. Geom., \textbf{ 11} (1976), 573--598. \bibitem {BT} M.Badiale, G. Tarantello; \emph{A Sobolev-Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics}, Arch. Ration. Mech. Anal., \textbf{163} (2002), 259--293. \bibitem{CFMS} D. Castorina, I. Fabbri, G. Mancini, K. Sandeep; \emph{Hardy-Sobolev inequilities and hyperbolic symmetry}, Rend. Lincei Mat. Appl., \textbf{19} (2008), 189--197. \bibitem{CFMS2} D. Castorina, I. Fabbri, G. Mancini, K. Sandeep; \emph{Hardy-Sobolev extremals, hyperbolic symmetry and scalar curvature equations}, J. Diffe. Equa., \textbf{246} (2009), 1187--1206. \bibitem{FabbriPHD} I. Fabbri; \emph{Remarks on some weighted Sobolev inequalities and applications}, PHD thesis, Univ. degli Studi Roma Tre, 2005. \bibitem{FMS} I. Fabbri, G. Mancini, K. Sandeep; \emph{Classification of solutions of a critical Hardy Sobolev operator}, Jour. Diff. Equa., \textbf{ 224} (2006), 258--276. \bibitem{GGMT} H. Grosse, V. Glaser, A. Martin, W. Thirring; \emph{A family of optimal conditions for the absence of bound states in a potential}, University Press, Princeton, 1976. \bibitem{JL} D. Jerison, J. Lee; \emph{The Yamabe problem on CR manifolds}, J. Diffe. Geom., \textbf{25} (1987), 167--197. \bibitem{Lieb} E. Lieb; \emph{Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities}, Ann. of Math., \textbf{118} (1983), 349--373. \bibitem{MS} G. Mancini and K. Sandeep; \emph{On a semilinear elliptic equation in $H^n$}, Annal Della Scuola Normale Superiore Di Pisa, \textbf{4} (2008), 635--671. \bibitem{P}A. Pankov; \emph{Periodic Nonlinear SchrĄ§odinger equation with application to photonic crystals}, Milan J. Math., \textbf{73} (2005), 259--287. \bibitem{S} S. Secchi; \emph{The Brezis-Nirenberg problem for the H\'{e}non equation: ground state solutions}, Adv. Nonli. Stud., \textbf{12} (2012), 383--394 \bibitem{SW} A. Szulkin, T. Weth; \emph{Ground state solutions for some indefinite variational problems}, J. Funct. Anal., \textbf{257} (2009), 3802--3822. \bibitem{SWW} A. Szulkin, T. Weth, M. Willem; \emph{Ground state solutions for a semilinear problem with critical exponent}, Differential Integral Equations, \textbf{22}(2009), 913--926. \bibitem{Talenti} G. Talenti; \emph{Best constant in Sobolev inequality}, Ann. Mat. Pura Appl., \textbf{110} (1976), 353--372. \end{thebibliography} \end{document}