\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 208, 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} \thanks{\copyright 2013 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2013/208\hfil Existence and asymptotic behavior] {Existence and asymptotic behavior of solutions for H\'enon equations in hyperbolic spaces} \author[H. He, W. Wang \hfil EJDE-2013/208\hfilneg] {Haiyang He, Wei Wang} % in alphabetical order \address{Haiyang He \newline College of Mathematics and Computer Science, Key Laboratory of High Performance Computing and Stochastic Information Processing, Ministry of Education, Hunan Normal University, Changsha, Hunan 410081, China} \email{hehy917@hotmail.com} \address{Wei Wang \newline College of Mathematics and Computer Science, Key Laboratory of High Performance Computing and Stochastic Information Processing, Ministry of Education, Hunan Normal University, Changsha, Hunan 410081, China} \email{ww224182255@sina.com} \thanks{Submitted May 29, 2013. Published September 19, 2013.} \subjclass[2000]{35J20, 35J60} \keywords{H\'enon equation; hyperbolic space; asymptotic behavior; blow up} \begin{abstract} In this article, we consider the existence and asymptotic behavior of solutions for the H\'{e}non equation \begin{gather*} -\Delta_{\mathbb{B}^N}u=(d(x))^{\alpha}|u|^{p-2}u, \quad x\in \Omega\\ u=0 \quad x\in \partial \Omega, \end{gather*} where $\Delta_{\mathbb{B}^N}$ denotes the Laplace Beltrami operator on the disc model of the Hyperbolic space $\mathbb{B}^N$, $d(x)=d_{\mathbb{B}^N}(0,x)$, $\Omega \subset \mathbb{B}^N$ is geodesic ball with radius $1$, $\alpha>0, N\geq 3$. We study the existence of hyperbolic symmetric solutions when $20, N\geq 3$. When posed in Euclidean space $\mathbb{R}^N$, problem \eqref{eq:1.1} becomes $$\label{eq:1.2} \begin{gathered} -\Delta u=|x|^{\alpha}|u|^{p-2}u, \quad x\in \Omega\\ u=0 \quad x\in \partial \Omega, \end{gathered}$$ where $\Omega$ is the unit ball in $\mathbb{R}^N$ with $N\geq 3$, $\alpha>0$ and $p >2$, which stems from the study of rotating stellar structures and is called H\'{e}non equation \cite{H}. Such a problem has been extensively studied, see for instance \cite{CP,N,S} etc. Interesting phenomenon concerning with problem \eqref{eq:1.2} was revealed recently that the exponent $\alpha$ affects the critical exponent for the existence of solutions. Precisely, it was shown in \cite{N} that for $p\in( 2, \frac{2N+2\alpha}{N-2})$, problem \eqref{eq:1.2} admits at least one radial solution. One also notices that the moving plane method in \cite{GNN} can not be applied to \eqref{eq:1.2} since the weight function $r\mapsto r^\alpha$ is increasing. So it can be expected that problem \eqref{eq:1.2} possesses non-radial solutions. Such solutions were found in \cite{SSW} for $2 < p <\frac{2N}{N-2}$ and in \cite{S} for $p = \frac{2N}{N-2}$. Furthermore, in \cite{CP}, the limiting behavior of the ground state solutions of \eqref{eq:1.2} was considered as $p\to 2^*=\frac {2N}{N-2}$. The authors showed that the maximum point of ground state solutions of \eqref{eq:1.2} concentrate on a boundary point of the domain as $p\to 2^*$. In their arguments, one of the key ingredients is to show that the ground state solutions $\{u_p\}$, $20$, problem \eqref{eq:1.1} possesses a ground solution $u_p$ which belongs to $H_0^1(\mathbb{B}^N)$ when $p\in (2,\frac{2N}{N-2})$. Moreover, there is a hyperbolic symmetry positive solution $u_p^{rad}$ for problem \eqref{eq:1.1} provided that $p\in (2,\frac{2N+2\alpha}{N-2})$. \end{theorem} \begin{theorem}\label{thm:1.2} Suppose $p\in(2,2^{*}),\alpha>0$, then the ground state solution $u_p$ satisfies (after passing to subsequence) for some $x_{0}\in\partial \Omega$, \begin{itemize} \item[(i)] $|\nabla_{\mathbb{B}^N} u_p|^2\to \mu\delta_{x_{0}}$ as $p\to 2^{*}$ in the sense of measure. \item[(ii)] $|u_p|^{2^{*}}\to \nu\delta_{x_{0}}$ as $p\to 2^{*}$ in the sense of measure, \end{itemize} where $\mu>0$, $\nu>0$ satisfy $\mu\geq S \nu^{2/2^*}$, $\delta_x$ is the Dirac mass at $x$. \end{theorem} \begin{theorem}\label{thm:1.3} Let $u_p$ be as in Theorem \ref{thm:1.2} and $x_p\in \bar{\Omega}$ be such $M_p'=u_p(x_p)=max_{x\in \bar{\Omega}}u_p(x)$, $\lambda'_p={M'_p}^{-2/(N-2)}$. Then, as $p\to 2^{*},M'_p\to +\infty$ and \begin{itemize} \item[(i)] $x_p$ is unique when $p$ close to $2^{*}$. Moreover, as $p\to 2^{*},dist_{\mathbb{B}^N}(x_p,\partial \Omega)\to 0$, $\operatorname{dist}_{\mathbb{B}^N}(x_p,\partial \Omega)/\lambda'_p \to \infty$, \item[(ii)] $\lim_{p\to 2^{*}}\int_{\Omega}|\nabla_{\mathbb{B}^N} (u_p-(\frac{1-|x|^2}{2})^\frac{N-2}{2}U_{\lambda_p,x_p})|^2 \,dV_{\mathbb{B}^N}=0$, where $\lambda_p$ is defined in Section 4. \end{itemize} \end{theorem} This paper is organized as follows. In section 2, we give some basic facts about hyperbolic space and the proof of Theorem \ref{thm:1.1}. In section 3, we show that $u_p$ is a minimizing sequence of $S$ as $p\to 2^*$, and then prove Theorem \ref{thm:1.2} by the concentration compactness principle. In section 4, we prove Theorem \ref{thm:1.3} mainly by a blow-up technique. \section{Preliminaries} Hyperbolic space $\mathbb{H}^N$ is a complete simple connected Riemannian manifold which has constant sectional curvature equal to $-1$. There are several models for $\mathbb{H}^N$ and we will use the Poincar\'{e} ball model $\mathbb{B}^N$ in this article. The Poincar\'{e} ball model for the hyperbolic space is $\mathbb{B}^N=\{x=(x_1, x_2,\dots, x_n)\in \mathbb{R}^N| \ |x|<1\}$ endowed with Riemannian metric $g$ given by $g_{ij}=(p(x))^2\delta_{ij}$ where $p(x)=\frac{2}{1-|x|^2}$. We denote the hyperbolic volume by $dV_{\mathbb{B}^N}$ and is given by $dV_{\mathbb{B}^N}=(p(x))^N \,dx$. The hyperbolic gradient and the Laplace Beltrami operator are: $\Delta_{\mathbb{B}^N}=(p(x))^{-N}\operatorname{div}((p(x))^{N-2} \nabla u)),\quad \nabla_{\mathbb{B}^N} u=\frac{\nabla u}{p(x)}$ where $\nabla$ and div denote the Euclidean gradient and divergence in $\mathbb{R}^N$, respectively. The hyperbolic distance $d_{\mathbb{B}^N}(x,y)$ between $x, y\in \mathbb{B}^N$ in the Poincar\'{e} ball model is given by the formula: $$\label{1.3} d_{\mathbb{B}^N}(x,y) =\operatorname{Arccosh}(1+\frac{2|x-y|^2}{(1-|x|^2)(1-|y|^2)}).$$ From this we immediately obtain for $x\in \mathbb{B}^N$, $d(x)=d_{\mathbb{B}^N}(0,x)=\log(\frac{1+|x|}{1-|x|}).$ Let us denote the energy functional corresponding to \eqref{eq:1.1} by $$\label{eq:2.1} I(u)= \frac{1}{2}\int_{\Omega}|\nabla_{\mathbb{B}^N}u|^2dV_{\mathbb{B}^N}- \frac{1}{p}\int_{\Omega}|d(x)|^{\alpha}|u|^pdV_{\mathbb{B}^N}$$ defined on $H_0^1(\Omega)$, where $H_0^{1}(\Omega)$ is the Sobolev space on $\mathbb{B}^N$ with the above metric $g$. We see that $u\in H_0^1(\Omega)$ is a solution of problem \eqref{eq:1.1} if and only if $v=(\frac{2}{1-|x|^2})^{\frac{N-2}{2}}u$ solves the following equation $$\label{eq:2.1'} -\Delta{v}+\frac{N(N-2)}{4}(\frac{2}{1-|x|^2})^2 v=({\ln{\frac{1+|x|}{1-|x|}}})^{\alpha}(\frac{1-|x|^2}{2}) ^{\frac{(N-2)p-2N}{2}}|v|^{p-2}v,$$ for $v\in H_{0}^{1}(\Omega')$, where $\Omega'$ is a ball in $\mathbb{R}^N$ centered at origin with radius $r=(e-1)/(e+1)$, $\alpha>0, p>2$. Let us define the energy functional corresponding to \eqref{eq:2.1'} by \label{eq:2.3'} \begin{aligned} J(v) &=\frac{1}{2} \int_{\Omega'} |\nabla v|^2+\frac{N(N-2)}{4}) (\frac{2}{1-|x|^2})^2 v^2\,dx\\ &\quad -\frac{1}{p}\int_{\Omega'}(\ln\frac{1+|x|}{1-|x|})^{\alpha} (\frac{1-|x|^2}{2})^{\frac{(N-2)p-2N}{2}}|v|^p \,dx. \end{aligned} Thus for any $u\in H_0^1(\Omega)$ if $\tilde{u}$ is defined as $\tilde{u}=(\frac{2}{1-|x|^2})^\frac{N-2}{2} u$, then $I(u)=J(\tilde{u})$. Moreover $\langle I'(u), v\rangle=\langle J'(\tilde{u}), \tilde{v}\rangle$ where $\tilde{v}$ is defined in the same way. \begin{proof}[Proof Theorem \ref{thm:1.1}] As $\ln\frac{1+|x|}{1-|x|}\leq \frac{2|x|}{1-|x|^2}$, $|x|\leq\frac{e-1}{e+1}$ and $\frac{2e}{(e+1)^2}\leq \frac{1-|x|^2}{2}\leq \frac{1}{2}$, firstly, for $\alpha\geq0,20, p>2$. \begin{lemma}\label{lm:3.1} The solution of \eqref{eq:3.1} satisfies $$\label{3.2} \begin{split} &\frac{\int_{\Omega'}(|\nabla v_p|^2 +\frac{N(N-2)}{4}(\frac{2}{1-|x|^2})^2|v_p|^2)dx} {(\int_{\Omega'}|v_p|^pdx)^{2/p}}\\ &\geq\frac{\int_{\Omega'}(|\nabla v_p|^2 +\frac{N(N-2)}{4}(\frac{2}{1-|x|^2})^2|v_p|^2)dx} {(\int_{\Omega'}|v_p|^{2^{*}}dx)^{2/2^*}}+O_{(2^{*}-p)}(1) \end{split}$$ for $p$ near $2^{*}$. \end{lemma} \begin{proof} By H\"{o}lder inequality $$\Big(\int_{\Omega'}|v_p|^pdx\Big)^{1/p} \leq(\int_{\Omega'}|v_p|^{2^{*}}dx)^{1/2^*} \Big(\operatorname{meas}\Omega'\Big)^{\frac{2^{*}-p}{2^{*}p}},$$ then Lemma \ref{lm:3.1} follows immediately. \end{proof} For $\varepsilon>0$ small enough, let $x_{0}=(\frac{e-1}{e+1}-\frac{1}{|\ln \varepsilon|},0,\dots ,0)\in R^N$, $$U_{\varepsilon}(x)=\frac{1}{(\varepsilon+|x-x_{0}|^2)^{\frac{N-2}{2}}},$$ $\varphi\in C_{0}^{\infty}(\Omega)$ be a cut-off function satisfying $\varphi(x)=\begin{cases} 1, & x\in B(x_{0},\frac{1}{2|\ln\varepsilon|})\\ 0, & x\in R^{n}\backslash B(x_{0},\frac{1}{|\ln\varepsilon|}) \end{cases}$ and $0\leq\varphi(x)\leq1$, $|\nabla\varphi(x)|\leq C|\ln\varepsilon|$ for all $x\in \mathbb{R}^N$, where $C$ is independent of $\varepsilon$, $B(x,r)$ denotes a ball centered $x$ with radius $r$. Set $v_{\varepsilon}=\varphi U_{\varepsilon}$, then $v_{\varepsilon}\in H_{0}^{1}(\Omega')$. \begin{lemma}\label{lm:3.1'} Let $v_{\varepsilon}$ be defined as above, then $\lim_{\varepsilon\to 0}\lim_{p\to 2^{*}} \frac{\int_{\Omega'}\big(|\nabla v_{\varepsilon}|^{2} +\frac{N(N-2)}{4}(\frac{2}{1-|x|^{2}})^2|v_{\varepsilon}|^{2}\big)dx} {\big(\int_{\Omega'}(\ln\frac{1+|x|}{1-|x|})^{\alpha} (\frac{1-|x|^{2}}{2})^{\frac{(N-2)p-2N}{2}}|v_{\varepsilon}|^{p}dx \big)^{2/p}}=S\,.$ \end{lemma} \begin{proof} On the one hand, from \cite{CP}\cite{S1}, we have $$\label{3.3} |v_{\varepsilon}|_p^2=|U|_p^2\varepsilon^{\frac{N}{p}-(N-2)}+CK_{1} (\varepsilon)|U|_p^{2-p}\varepsilon^{\frac{(N-2)p}{2}-\frac{N}{2} +\frac{N}{p}-(N-2)},$$ $$\label{3.4} |\nabla v_{\varepsilon}|^2_{2}=|\nabla U|_{2}^2 \varepsilon^{-\frac{(N-2)}{2}}+ \begin{cases} C|\ln \varepsilon|^{N-2}+o(|\ln\varepsilon|^{N-2}), &N\geq5,\\ C|\ln\varepsilon|^2(\ln(2|2\ln\varepsilon|))+O(|\ln\varepsilon|^2), & N=4,\\ C|\ln\varepsilon|^2, & N=3, \end{cases}$$ and $$\label{3.5} |v_{\varepsilon}|^2_{2}=O(\frac{1}{|\ln\varepsilon|^2}).$$ On the other hand, we have \begin{align*} &\int_{\Omega'}(\ln\frac{1+|x|}{1-|x|})^{\alpha} (\frac{1-|x|^2}{2})^{\frac{(N-2)p-2N}{2}}|v_{\varepsilon}|^pdx\\ &\geq(\ln\frac{e-\frac{e+1}{|\ln\varepsilon|}}{1+\frac{e+1} {|\ln\varepsilon|}})^{\alpha} (\frac{1}{2})^{\frac{(N-2)p-2N}{2}} \int_{\Omega'}|v_{\varepsilon}|^p dx, \end{align*} and $\int_{\Omega'}\frac{N(N-2)}{4}(\frac{2}{1-|x|^2})^2|v_{\varepsilon}|^{ 2} \,dx \leq \frac{N(N-2)}{4}\frac{(e+1)^2}{2e}\int_{\Omega'} |v_{\varepsilon}|^{2} \,dx.$ By\eqref{3.3}--\eqref{3.5}, for $N\geq5$, we have \begin{align} &\lim_{\varepsilon\to 0}\lim_{p\to 2^{*}} \frac{\int_{\Omega'}(\nabla v_{\varepsilon}|^2+\frac{N(N-2)}{4} \frac{2}{1-|x|^2}|v_{\varepsilon}|^{2})dx} {(\int_{\Omega'}(\ln\frac{1+|x|}{1-|x|})^{\alpha} (\frac{1-|x|^2}{2})^{\frac{(N-2)p-2N}{2}}|v_{\varepsilon}|^pdx)^{2/p}} \nonumber \\ &\leq \lim_{\varepsilon\to 0}\lim_{p\to 2^{*}} \frac{1}{(\ln\frac{e-\frac{e+1}{|\ln\varepsilon|}} {1+\frac{e+1}{|\ln\varepsilon|}})^{\frac{2\alpha}{p}}} \times\frac{1}{(\frac{1}{2})^{\frac{(N-2)p-2N}{2}}} \nonumber\\ &\quad\times\frac{|\nabla U|_{2}^2 \varepsilon^{-\frac{N-2}{2}} +C|\ln\varepsilon|^{N-2}+o( |\ln\varepsilon|^{N-2}) +O(\frac{1}{|\ln\varepsilon|^2})} {|U|_p^2\varepsilon^{\frac{N}{p}-(N-2)} +C K_{1}(\varepsilon)|U|_p^{2-p}\varepsilon^{\frac{(N-2)p}{2} -\frac{N}{2}+\frac{N}{p}-(N-2)}} \nonumber\\ &=\lim_{\varepsilon\to 0}(\ln\frac{e-\frac{e+1}{|\ln\varepsilon|}} {1+\frac{e+1}{|\ln\varepsilon|}})^{\frac{2\alpha}{2^{*}}} \times\frac{|\nabla U|_{2}^2\varepsilon^{-\frac{N-2}{2}} +C|\ln\varepsilon|^{N-2}+o(|\ln\varepsilon|^{N-2}) +O(\frac{1}{|\ln\varepsilon|^2})}{|U|_{2}^{2^{*}}\varepsilon^{-\frac{N-2}{2} }+C |\ln\varepsilon|^N\varepsilon} \nonumber\\ &=\lim_{\varepsilon\to 0}(\ln\frac{e-\frac{e+1}{|\ln\varepsilon|}} {1+\frac{e+1}{|\ln\varepsilon|}})^{\frac{2\alpha}{2^{*}}} \times \frac{|\nabla U|_{2}^2 +C(\varepsilon^{\frac{1}{2}}|\ln\varepsilon|)^{N-2}}{|U|_{2^{*}}^2 +C(\varepsilon^{\frac{1}{2}} |\ln\varepsilon|)^{N-2}} =\frac{|\nabla U|_{2}^2}{|U|_{2^{*}}^2}. \label{3.6} \end{align} Moreover, $$\frac{\int_{\Omega'}\big(|\nabla v_{\varepsilon}|^2 +\frac{N(N-2)}{4}\frac{2}{1-|x|^2}|v_{\varepsilon}|^{ 2}\big)dx} {\big(\int_{\Omega'}(\ln\frac{1+|x|}{1-|x|})^{\alpha} (\frac{1-|x|^2}{2})^{\frac{(N-2)p-2N}{2}}|v_{\varepsilon}|^pdx\big)^{2/p}} \geq \frac{\int_{\Omega'} \big(|\nabla v_{\varepsilon}|^2+\frac{N(N-2)}{4}\frac{2} {1-|x|^2}|v_{\varepsilon}|^{2}\big)dx} {(\frac{e}{(1+e)^2})^{\frac{(N-1)p-2N}{p}} \big(\int_{\Omega'}|v_{\varepsilon}|^pdx\big)^{2/p}}.$$ Similarly, we have $$\label{3.7} \lim_{\varepsilon\to 0}\lim_{p\to 2^{*}} \frac{\int_{\Omega}(|\nabla v_{\varepsilon}|^2 +\frac{N(N-2)}{4}\frac{2}{1-|x|^2}|v_{\varepsilon}|^{2})dx} {(\int_{\Omega}(\ln\frac{1+|x|}{1-|x|})^{\alpha} (\frac{1-|x|^2}{2}dx)^{\frac{(N-2)p-2N}{2}}|v_{\varepsilon}|^p)^{2/p}} \geq\frac{|\nabla U|_{2}^2}{|U|_{2^{*}}^2}.$$ Combining \eqref{3.6} and \eqref{3.7}, we can complete the proof for $N\geq5$. The case $N=3,4$ can been proved similarly. \end{proof} \begin{lemma}\label{lm:3.1''} There holds \begin{gather}\label{3.8} \lim_{p\to 2^{*}}\frac{\int_{\Omega'}(|\nabla v_p|^2 +\frac{N(N-2)}{4}(\frac{2}{1-|x|^2})^2|v_p|^2)dx}{(\int_{\Omega'} (\ln\frac{1+|x|}{1-|x|})^{\alpha}(\frac{1-|x|^2}{2}) ^{\frac{(N-2)(p-1)-N-2}{2}}|v_p|^pdx)^{2/p}}= S, \\ \label{3.9} \lim_{p\to 2^{*}}\frac{\int_{\Omega'}|\nabla v_p|^2dx}{(\int_{\Omega'} |v_p|^pdx)^{2/p}}=\lim_{p\to 2^{*}} \frac{\int_{\Omega'}(|\nabla v_p|^2+\frac{N(N-2)}{4} (\frac{2}{1-|x|^2})^2|v_p|^2)dx}{(\int_{\Omega'} |v_p|^pdx)^{2/p}}= S. \end{gather} \end{lemma} \begin{proof} By the definition of $\{v_p\}$ and Lemma \ref{lm:3.1'}, noting $\ln\frac{1+|x|}{1-|x|}\leq1$, $\frac{2e}{(e+1)^2}\leq \frac{1-|x|^2}{2}\leq \frac{1}{2}$, and $(\frac{1-|x|^2}{2})^{\frac{(N-2)p-2N}{2}}\to 1$ as $p\to 2^{*}$, we have $$\label{3.10} \begin{split} &\frac{\int_{\Omega'}(|\nabla v_p|^2+\frac{N(N-2)}{4} (\frac{2}{1-|x|^2})^2|v_p|^2)dx}{(\int_{\Omega'}|v_p|^pdx)^{2/p}}\\ &\leq\frac{\int_{\Omega'}(|\nabla v_p|^2+\frac{N(N-2)}{4}(\frac{2}{1-|x|^2})^2|v_p|^2)dx}{(\int_{\Omega'}(\ln\frac{1+|x|}{1-|x|})^{\alpha} (\frac{1-|x|^2}{2})^{\frac{(N-2)p-2N}{2}}|v_p|^pdx)^{2/p}}\\ &\leq\frac{\int_{\Omega'}(|\nabla v_{\varepsilon}|^2+\frac{N(N-2)}{4} (\frac{2}{1-|x|^2})^2|v_{\varepsilon}|^2)dx}{(\int_{\Omega'} (\ln\frac{1+|x|}{1-|x|})^{\alpha}(\frac{1-|x|^2}{2}) ^{\frac{(N-2)p-2N}{2}}|v_{\varepsilon}|^pdx)^{2/p}} =S+o(\varepsilon). \end{split}$$ In addition, for any $p$, $20$. As \cite{CP}, using the concentration-compactness principle, we can prove that the ground state solution $v_p$ of problem \eqref{eq:3.1} satisfies (after passing to subsequence) for some $x_{0}\in\partial \Omega'$, \begin{itemize} \item[(i)] $|\nabla v_p|^2\rightharpoonup\mu\delta_{x_{0}}$ as $p\to 2^{*}$ in the sense of measure. \item[(ii)] $|v_p|^{2^{*}}\rightharpoonup\nu\delta_{x_{0}}$ as $p\to 2^{*}$ in the sense of measure, \end{itemize} where $\mu>0$, $\nu>0$ satisfy $\mu\geq S \nu^{2/2^*}$, $\delta_x$ is the Dirac mass at $x$. Given that $\frac{2e}{(e+1)^2}\leq \frac{1-|x|^2}{2}\leq \frac{1}{2}$, $u_p=(\frac{1-|x|^2}{2})^\frac{N-2}{2}v_p$ and $v_p\rightharpoonup 0$ in $H_0^1(\Omega')$, we have \begin{itemize} \item[(i)] $|\nabla_{\mathbb{B}^N} u_p|^2\rightharpoonup\mu\delta_{x_{0}}$ as $p\to 2^{*}$ in the sense of measure. \item[(ii)] $|u_p|^{2^{*}}\rightharpoonup\nu\delta_{x_{0}}$ as $p\to 2^{*}$ in the sense of measure, \end{itemize} where $\mu>0$, $\nu>0$ satisfy $\mu\geq S \nu^{2/2^*}$, $\delta_x$ is the Dirac mass at $x$. \end{proof} \section{Proof of Theorem \ref{thm:1.3}} In this section, we shall study the asymptotic of the ground state solution and prove Theorem \ref{thm:1.3}. Set $$M_p=\sup_{x\in\bar{\Omega'}}v_p(x)=v_p(x_p),x_p\in \bar{\Omega'}.$$ \begin{proposition}\label{pro:4.1} $M_p\to +\infty$ as $p\to 2^{*}$. \end{proposition} \begin{proof} We need only to prove this proposition for any subsequence ${p_{k}}$, such that $p_{k}\to 2^{*}$ as $k\to +\infty$. Assume by contradiction that there exists a positive constant $c$ such that $M_{p_{k}}\leq c$ for all $k$. For Theorem \ref{thm:1.2}, $v_{p_{k}}\to 0$ a.e. $\Omega'$. By Fatou's Lemma, Egoroff Theorem and the fact that $\int_{\Omega'}|v_{p_{k}}|^{2^{*}}=1$, we have $u_{p_{k}}\to 0$ weakly in $L^{2^{*}}(\Omega')$. So, for $\sigma>0$ small, due to the compactness of $L^{2^{*}}(\Omega')\hookrightarrow L^{2^{*}-\sigma}(\Omega')$, we have a subsequence(still denoted by $\{v_{p_{k}}\}$) such that $$1=\int_{\Omega'}|v_{p_{k}}|^{2^{*}}dx \leq|v_{p_{k}}|^{\sigma}_{L^{\infty}(\Omega')} \int_{\Omega'}|v_{p_{k}}|^{2^{*}-\sigma}dx\leq c^{\sigma}\int_{\Omega'} |v_{p_{k}}|^{2^{*}-\sigma}dx\to 0$$ as $k\to \infty$, which is impossible. \end{proof} \textbf{Proof of Theorem \ref{thm:1.3}} We follow the blow up technique used by Gidas and Spruck in\cite{GS}. Suppose that for a subsequence of $p$ as $p\to 2^{*}, x_p\to x_{0}\in \bar{\Omega'}$. Let $\lambda_p$ be a sequence of positive numbers defined by $\lambda_p^{\frac{N-2}{2}}M_p=1$ and $y=\frac{x-x_p}{\lambda_p}$. Define the scaled function $$\label{4.1} w_p(y)=\lambda_p^{\frac{N-2}{2}}v_p(x)$$ and the domain $$\label{4.2} \Omega'_p=\{y\in R^N:\lambda_py+x_p\in\Omega'\}.$$ Since $M_p\to +\infty$, we have $\lambda_p\to 0$ as $p\to 2^{*}$. It is easy to see that $w_p(y)$ satisfies \label{4.3} \begin{gathered} \begin{aligned} &-\Delta w_p+\frac{N(N-2)}{4}(\frac{2\lambda_p}{1-|\lambda_py+x_p|^2})^2w_p\\ &=(\ln\frac{1+|y\lambda_p+x_p|}{1-|y_p+x_p|})^{\alpha} \lambda_p^{\frac{(N-2)(2^{*}-p)}{2}}(\frac{1-|\lambda_py+x_p|^2}{2}) ^{\frac{(N-2)p-2N}{2}}w^{p-1}_p, \quad y\in\Omega'_p, \end{aligned}\\ w_p=0,\quad y\in\partial\Omega'_p,\\ 0n)$. Choosing$p$sufficiently close to$2^{*}$, we obtain by Morrey's theorem that$\|w_p\|_{C^{1,\theta}}(B(0,l))(0<\theta<1)$is also uniformly bounded. It follows that for any sequence$p\to 2^{*}$, there exists a subsequence$p_{k}\to 2^{*}$such that$w_{p_{k}}\to w$in$W^{2,r}\cap C^{1,\theta}(r>N)$on$B(0,l)$. By H\"older continuity$v(0)=1$. Furthermore, since for$y\in B(0,l)$, $$\lambda_{p_{k}}y+x_{p_{k}}\to x_{0} \quad\text{as } k\to +\infty,$$ as in \cite{GS} we can also prove that$w$is well defined in all$R^N$and$w_{p_{k}} \to w$in$W^{2,r}\cap C^{1,\theta}(r>N)$on any compact subset. Therefore$w(y)$is a solution of $$\label{4.9} -\Delta w=(\ln\frac{1+|x_{0}|}{1-|x_{0}|})^{\alpha}L(2^{*})w^{2^{*}-1}.$$ If$L(2^{*})=0$or$x_{0}=0$, then$-\Delta w=0$in$R^N$. Thus$w\equiv0$, which is impossible since$w(0)=1$. If$L(2^{*})\in(0,1]$, then by \eqref{4.9}, Equation \eqref{4.3} is \label{4.10} \begin{gathered} -\Delta w=cw^{2^{*}-1},\quad y\in R^N\\ w\to 0 \quad\text{as }|y|\to \infty\\ 0-\frac{d_p}{\lambda_p}\}$ for some small $\delta>0$ and satisfies \eqref{4.3}. Moreover, $\sup w_p(y)=w_p(0)=1$. We assert that \begin{itemize} \item[(I)] $\frac{d_p}{\lambda_p}\to +\infty$ as $p\to 2^{*}$; \item[(II)] $L(2^{*})=1$. \end{itemize} Proof of (I). Assume to the contrary that $\frac{d_p}{\lambda_p}$ is uniformly bounded from above, and (by going to a subsequence if necessary) $\frac{d_p}{\lambda_p}\to s$ with $s\geq0$. Repeating the compactness argument as in the case (1), noting that $|x_{0}|=\frac{e-1}{e+1}$, we get a subsequence of $w_p$ converging to $w(y)$ satisfying \label{4.8} \begin{gathered} -\Delta w=L(2^{*})w^{2^{*}-1},\quad y\in R_{s}^N =\{y=(y_{1},\dots ,y_{n-1},y_{N}):y_{N}\geq-s\}\\ w=0,\quad y\in\partial R^N_{s},\\ 00)\},\\ w(y)=0,\quad y\in \partial R^N_{+} \end{gathered} has a unique solution $w=0$, we conclude that \eqref{4.8} possesses a unique trivial solution 0 for any case of $L(2^{*})$, which contradicts $w(0)=1$. So we can have only $\frac{d_p}{\lambda_p}\to +\infty$ as $p\to 2^{*}$. Proof of (II). Assertion(I) implies $\Omega'_p\to \Omega'_{2^{*}}=R^N$. Similarly by the above regularity theorems in the theory of elliptic equation and $|x_{0}|=\frac{e-1}{e+1}$, we obtain a subsequence of $w_p$ converging to some function $w(y)$ satisfying \label{eq:4.14} \begin{gathered} -\Delta w=L(2^{*})w^{2^{*}-1},\quad y\in R^N,\\ w(y)\to 0,\quad |y|\to \infty,\\ 00$,$y_{0}\in R^N$. Since$v$attains its maximum 1 at$y=0$, we have$\varepsilon=1$and$y_{0}=0$. Therefore$w=U$. Note that the limit of$ \{w_p\}$does not depend on the choice of subsequence by the uniqueness of$U$. Hence the whole sequence$\{w_p\}$must converge to$U$. Let$z_p=w_p-U$. Then$z_p\rightharpoonup0$weakly in$H^{1}(\Sigma)$for any bounded subset$\Sigma\subset R^N$, and $$\label{eq:4.15} \begin{gathered} -\Delta z_p+\frac{N(N-2)}{4}(\frac{2\lambda_p}{1-|\lambda_py+x_p|^2})^2w_p =Q_p(y)w_p^{p-1}-U^{2^{*}-1},\quad y\in\Omega'_p\\ z_p=-U,\quad y\in\partial\Omega'_p \end{gathered}$$ where $$Q_p(y)=(\ln\frac{1+|\lambda_py+x_p|}{1-|\lambda_py+x_p|})^{\alpha} (\frac{1-|\lambda_py+x_p|^2}{2})^{\frac{(N-2)p-2N}{2}} \lambda_p^{\frac{(N-2)(2^{*}-p)}{2}}.$$ Multiplying \eqref{eq:4.15} by$z_p$and integrating by parts, we obtain, as$p\to 2^{*}, \label{4.16} \begin{aligned} \int_{\Omega'_p}|\nabla z_p|^2dx &= \int_{\Omega'_p}[Q_p(y)w_p^{p-1}-U^{2^{*}-1}]z_p\\ &\quad -\int_{\Omega'_p}\frac{N(N-2)}{4} (\frac{2}{1-|\lambda_py+x_p|^2})^2w_pz_p +\int_{\partial\Omega'_p}\frac{\partial z_p}{\partial \nu}U ds\\ &= \int_{\Omega'_p}Q_p(y)|z_p|^p+o_{(2^{*}-p)}(1). \end{aligned} The last equality follows from the weak convergence ofw_p$in$H^{1}(\Sigma)$and the decay of$U$at infinity. As$p\to 2^{*}$, $$\label{eq:4.14'} \int_{\Omega'_p}|\nabla z_p|^2\geq S \Big(\int_{\Omega'_p}Q_p(y)|z_p|^p\Big)^{2/p}+o_{2^{*}-p}(1)$$ If$\int_{\Omega'_p}|\nabla z_p|^2dx\to \rho>0, by \eqref{eq:4.14'}, we see easily that $$\int_{\Omega'_p}|\nabla z_p|^2=\int_{\Omega'_p}Q_p(y)|z_p|^pdx+o_{2^{*}-p}(1) \geq S^{N/2}+o_{2^{*}-p}(1)\ \quad\text{as } p\to 2^{*}.$$ Then by \eqref{eq:2.1'} and Corollary \ref{cor:3.1}, we have $$\label{4.17} J(v_p)=\frac{1}{N}S^{N/2}+o_{(2^{*}-p)}(1)\ as \ p\to 2^{*}.$$ On the other hand, as we done in obtaining \eqref{4.16}, \begin{align*} J(v_p) &=\frac{1}{2}\int_{\Omega'_p}|\nabla U|^2 -\frac{1}{p}\int_{\Omega'_p}(\frac{2}{1-|\lambda_py+x_p|^2}) ^{\frac{(N-2)p-2N}{2}}U^p\\ &\quad +\frac{1}{2}\int_{\Omega'_p}|\nabla w_p|^2 -\frac{1}{p}\int_{\Omega'_p}Q_p(y)(\frac{2}{1-|\lambda_py+x_p|^2}) ^{\frac{(N-2)p-2N}{2}}w_p^p\\ &\quad +\frac{N(N-2)}{4}\int_{\Omega'_p}(\frac{2\lambda_p}{1-|\lambda_py+x_p|^2}) ^2w_p^2\\ &\quad +\frac{N(N-2)}{4}\int_{\Omega'_p} (\frac{2\lambda_p}{1-|\lambda_py+x_p|^2})^2U^2+o_{(2^{*}-p)}(1) \\ &=\frac{1}{2}\int_{R^N}|\nabla U|^2-\frac{1}{2^{*}}\int_{\Omega'_p}U^{2^{*}} +\frac{1}{2}\int_{\Omega'_p}|\nabla w_p|^2 -\frac{1}{p}\int_{\Omega'_p}Q_p(y)w^p_p+o_{2^{*}-p}(1)\\ &\geq \frac{2}{N}S^{N/2}+o_{(2^{*}-p)}(1) \end{align*} which contradicts \eqref{4.17}. Thus\rho=0$, and we obtain $$\label{4.18} \lim_{p\to 2^{*}}\int_{\Omega'}|\nabla(v_p-U_{\lambda_p,x_p})|^2=0.$$ Since $$\frac{2e}{(e+1)^2}\leq \frac{1-|x|^2}{2}\leq \frac{1}{2}, \quad u_p=(\frac{1-|x|^2}{2})^\frac{N-2}{2}v_p,$$ part (ii) of Theorem \ref{thm:1.3} is proved. To complete our proof of Theorem \ref{thm:1.3}, we need only to show that$x_p$is unique for$p$close to$2^{*}$. Suppose that this is not true, then exist$x_p^{i}$,$i=1,2$, such that$M_p=v_p(x_p^{i})$for$i=1,2$. For$x_p^{i}$by choosing subsequence as$p\to 2^{*}$, we have either $$\label{4.19} \frac{|x_p^{1}-x_p^2|}{\lambda_p}\to +\infty$$ or $$\label{4.20} \frac{|x_p^{1}-x_p^2|}{\lambda_p}\leq c<+\infty$$ where$c$is some positive constant independent of$p$. Suppose that \eqref{4.20} holds, then the scaled function$w_p$would have two local maximum points in$B(0,l)$for$l$large enough and$p$close to$2^{*}$. On the other hand, by \cite[Lemma 4.2]{NW} and by using the similar arguments to \cite{NW}, we can also verify that$w_p$has only one local maximum point. So we get a contradiction. Assume that \eqref{4.19} holds, then from \eqref{4.18} we obtain $$\label{4.21} \lim_{p\to 2^{*}}\int_{\Omega'}|\nabla(U_{\lambda_p, x_p^{1}} -U_{\lambda_p,x_p^2})|^2=0.$$ Setting$(\Omega')_p^{1}=\{y|\lambda_py+x_p^{1}\in \Omega\}$and$m_p=\frac{x_p^{1}-x_p^2}{\lambda_p}$, we have $$0=2S^{N/2}-2\lim_{p\to 2^{*}}\int_{(\Omega')_p^{1}} \nabla U\nabla U_{1,z_p}.$$ Since$|m_p|\to +\infty$, we obtain$\lim_{p\to 2^{*}} \int_{(\Omega')_p^{1}}\nabla U\nabla U_{1,z_p}=0$, this contradicts \eqref{4.21} and hence \eqref{4.19} does not hold, either. Since \begin{gather*} u_p=(\frac{1-|x|^2}{2})^\frac{N-2}{2}v_p,\quad \frac{2e}{(e+1)^2}\leq \frac{1-|x|^2}{2}\leq \frac{1}{2}, \\ M'_p=u_p(x_p)=\max_{x\in \bar{\Omega}}u_p(x)\,, \end{gather*} it follows that$M'_p\to +\infty$as$p\to 2^{*}$. 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