\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  $2<p<\frac{2N+2\alpha}{N-2}$.  We also investigate asymptotic
 behavior of the ground  state solution when $p$ tends to the critical exponent
 $2^* =\frac{2N}{N-2}$ with $N\geq 3$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks

\section{Introduction and main result}

In this paper, we consider the problem
 \begin{equation}\label{eq:1.1}
  \begin{gathered}
-\Delta_{\mathbb{B}^N}u=(d(x))^{\alpha}|u|^{p-2}u,  \quad  x\in   \Omega\\
u=0 \quad   x\in \partial \Omega,
\end{gathered}
\end{equation}
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 $1$, $\alpha>0, N\geq 3$.

 When posed in Euclidean space $\mathbb{R}^N$, problem \eqref{eq:1.1}
becomes
  \begin{equation}\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}
\end{equation}
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 $\{u_p\}$, $2<p<\frac{2N}{N-2}$, of problem
\eqref{eq:1.2} is actually a minimizing sequence of the problem
\[
S=\inf\Big\{\frac{\int_\Omega|\nabla u|^2\,dx}{(\int_\Omega
|u|^{2^*} \,dx)^{2/2^*}} : u\in H_0^1(\Omega), \, u\not\equiv
0\Big\}
\]
as $p\to 2^*$, and use the fact that $S$ is
attained in $\mathbb{R}^N$ by the instanton
$U=1/(1-|x|^2)^{(N-2)/2}$.

In the hyperbolic space, the existence of Brezis-Nirenberg problem
for the critical  equation
\begin{equation}\label{eq:1.4}
-\Delta_{\mathbb{B}^N}u=|u|^{2^{*}-2}u+\lambda u,  \quad
u\geq 0, u\in H_{0}^{1}(\Omega)
 \end{equation}
has been studied in \cite{S1} and  the results are very similar to the results
in the Euclidean case. However, for problem \eqref{eq:1.1}, there are
some difference from Euclidean space.
Firstly, the weight function $d(x)$ depends
on the Riemannian hyperbolic distance $r$ from a pole $o$. Secondly,
the main  purpose in this paper is to study the  profile of ground state
solution $u_p$ of problem \eqref{eq:1.1} as $p\to  2^*$,
in particular, the asymptotic
behavior of $u_p$ and the limit location of the maximum point of $u_p$
as $p\to  2^*$. In generally, in order to prove the ground state
solution is a minimizing sequence of the problem
\begin{equation}
 S_{\mathbb{B}^N}(\Omega)=\inf\Big\{\frac{\int_\Omega|\nabla_{\mathbb{B}^N} u|^2
\,dx  }{(\int_\Omega |u|^{2^*} \,dx)^{2/2^*}} : u\in H_0^1(\Omega), \,
u\not\equiv 0\Big\},
\end{equation}
one will use the unique positive solution of the problem
 \begin{equation}\label{eq:1.5}
 -\Delta_{\mathbb{B}^N}u=u^{2^*-1}\quad\text{in }\mathbb{B}^N.
  \end{equation}
However, in \cite{BS}, Mancini  and Sandeep proved that problem \eqref{eq:1.5}
did not have any positive solutions.

Motivated by above mentioned works,  we study problem \eqref{eq:1.1}
in this paper. Our main results are as follows.

\begin{theorem}\label{thm:1.1}
For $\alpha>0$, 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:
\begin{equation}\label{1.3}
d_{\mathbb{B}^N}(x,y)
=\operatorname{Arccosh}(1+\frac{2|x-y|^2}{(1-|x|^2)(1-|y|^2)}).
\end{equation}
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
\begin{equation}\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}
 \end{equation}
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
\begin{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,
\end{equation}
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
\begin{equation}\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}
\end{equation}
 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 $\alpha\geq0,2<p<2^{*}$, we have the variational problem
  \begin{equation}\label{2.2}
S_{\alpha,p}:=\inf_{0\not\equiv v\in H_{0}^{1}(\Omega')}
\frac{\int_{\Omega'}|\nabla v|^2+\frac{N(N-4)}{2}
(\frac{2}{1-|x|^2})^2v^2dx}{(\int_{\Omega'}
(\ln\frac{1+|x|}{1-|x|})^{\alpha}
(\frac{1-|x|^2}{2})^{\frac{(N-2)p-2N}{2}}|v|^pdx)^{2/p}}
\end{equation}
which is solved by a $v_p$.
Thus  $u_p=(\frac{2}{1-|x|^2})^{-\frac{N-2}{2}}v_p$ is a ground
state solution of \eqref{eq:1.1}.

Secondly,  by \cite{N}, we have $u\mapsto |x|^\frac{\alpha}{p} u$
from $H_r^1(\Omega')$ to $L^p(\Omega')$ is compact for
$p\in (2, \frac{2N}{N-2-\frac{2\alpha}{p}} )$$(2<p<\frac{2N+2\alpha}{N-2}) $.
Then the problem
\begin{equation}
S_{\alpha,p}^{R}:=\inf_{0\not\equiv v\in H_{0,rad}^{1}(\Omega')}
\frac{\int_{\Omega'}|\nabla v|^2+\frac{N(N-4)}{2}
(\frac{2}{1-|x|^2})^2v^2dx}{(\int_{\Omega'}
(\ln\frac{1+|x|}{1-|x|})^{\alpha}(\frac{1-|x|^2}{2})^{\frac{(N-2)p-2N}{2}}|v|^pdx)^{2/p}}
\end{equation}
is also attained by a $v^{rad}_p$,
where $H_{0,rad}^{1}(\Omega')$ denotes the subspace of radial functions
in $H_{0}^{1}(\Omega')$.
Then  $u^{rad}_p=(\frac{2}{1-|x|^2})^{-\frac{N-2}{2}}v^{rad}_p$
is a hyperbolic symmetry  solution of \eqref{eq:1.1}.
\end{proof}

 \section{Proof of Theorem \ref{thm:1.2}}

 Let us consider the  problem
\begin{equation} \label{eq:3.1}
\begin{gathered}
-\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,  \quad x\in\Omega'\\
v=0,\quad  x\in\partial\Omega'
\end{gathered}
\end{equation}
where  $\Omega'$ is a ball in $\mathbb{R}^N$ centered at
origin with radius $r=\frac{e-1}{e+1}$,  $\alpha>0, p>2$.

\begin{lemma}\label{lm:3.1}
The solution of \eqref{eq:3.1} satisfies
\begin{equation}\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}
 \end{equation}
 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
\begin{equation}\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)},
\end{equation}
\begin{equation}\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}
\end{equation}
and
\begin{equation}\label{3.5}
|v_{\varepsilon}|^2_{2}=O(\frac{1}{|\ln\varepsilon|^2}).
\end{equation}

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
\begin{equation}\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}.
\end{equation}
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
\begin{equation} \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}
\end{equation}
In addition, for any $p$, $2<p<2^{*}$,
\begin{equation}
S\leq\frac{\int_{\Omega'}|\nabla v_p|^2dx}{(\int_{\Omega'}|v_p|^pdx)^{2/p}}
\leq\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}}
\end{equation}
which combined with Lemma \ref{lm:3.1},  gives \eqref{3.8} and \eqref{3.9}.
\end{proof}

Lemma \ref{lm:3.1''} implies that $\{v_p\}$ is actually a
minimizing sequence of $S$. In fact, we have

\begin{corollary}\label{cor:3.1}
\[
\lim_{p\to 2^{*}}\int_{\Omega'}|\nabla v_p|^2 \,dx=S^\frac{N}{2}.
\]
\end{corollary}


\begin{corollary}\label{cor:3.2}
When $p=2^{*}$, Equation \eqref{eq:2.1'} does not possess any ground
state solutions.
\end{corollary}

\begin{proof}
Assume to the contrary that $S_{\alpha,2^{*}}$ can be achieved by
 $v_{2^{*}}\in H_{0}^{1}(\Omega')$, by Lemma \ref{lm:3.1''},
$S_{\alpha,2^{*}}=S_{0,2^{*}}=S$,  so
\begin{align*}
S_{\alpha,2^{*}}
&=\frac{\int_{\Omega'}(|\nabla v_{2^{*}}|^2
 +\frac{N(N-2)}{4}(\frac{2}{1-|x|^2})^2|v_{2^{*}}|^2)dx}
{(\int_{\Omega'}(\ln\frac{1+|x|}{1-|x|})^{\alpha}
|v_{2^{*}}|^{2^{*}}dx)^{2/2^*}}\\
&\geq \frac{\int_{\Omega'}(|\nabla v_{2^{*}}|^2
+\frac{N(N-2)}{4}(\frac{2}{1-|x|^2})^2|v_{2^{*}}|^2) \,dx}
{(\int_{\Omega'}|v_{2^{*}}|^{2^{*}}dx)^{2/2^*}}\\
&\geq \frac{\int_{\Omega'}|\nabla v_{2^{*}}|^2dx}
{(\int_{\Omega'}|v_{2^{*}}|^{2^{*}}\,dx)^{2/2^*}}\geq S.
\end{align*}
Hence $\frac{\int_{\Omega'}|\nabla v_{2^{*}}|^2\,dx}
{(\int_{\Omega'}|v_{2^{*}}|^{2^{*}}\,dx)^{2/2^*}}=S$,  which is impossible
since $ S $ cannot be achieved in a bounded domain.
\end{proof}


\begin{proof}[Proof  of Theorem \ref{thm:1.2}]
Suppose $p\in(2,2^{*}),\alpha>0$.  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
\begin{equation}\label{4.1}
w_p(y)=\lambda_p^{\frac{N-2}{2}}v_p(x)
\end{equation}
and the domain
\begin{equation}\label{4.2}
\Omega'_p=\{y\in R^N:\lambda_py+x_p\in\Omega'\}.
\end{equation}
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
\begin{equation}\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,\\
0<w_p\leq1,\quad w_p=1.
\end{gathered}
\end{equation}
 By Proposition \ref{pro:4.1}, we can have $M_p\geq1$ for $p$ close to $2^{*}$.
Therefore $0\leq\lambda_p\leq1$.
 Setting $L(p)=\lambda_p^{\frac{(N-2)(2^{*}-p)}{2}}, L(2^{*})
=\lim_{p\to 2^{*}}L(p)$, by choosing subsequence if necessary,
we have one of the three cases:
\begin{itemize}
\item[(i)] $L(2^{*})=0$;
\item[(ii)] $L(2^{*})=\beta\in(0,1)$;
\item[(iii)] $L(2^{*})=1$.
\end{itemize}
For the location of $x_{0}\in\bar{\Omega'}$, we also have one of the
two cases:
(1) $x_{0}\in\Omega'$,  and (2) $x_{0}\in\partial\Omega'$.


(1) Assume $x_{0}\in\Omega'$, let $2d$ denote the distance of
$x_{0}$ to $\partial\Omega'$. For $p$ close to $2^{*}$, $w_p(y)$
is well defined in the ball $B(0,\frac{d}{\lambda_p})$ and
\begin{gather*}
\sup_{y\in B(0,\frac{d}{\lambda_p})}w_p(y)=w_p(0)=1,\\
\Omega'_p\to \Omega'_{2^{*}}=R^N,as~~ p\to  2^{*},\\
B(0,\frac{d}{\lambda_p})\to  R^N,as~~ p\to  2^{*},\\
\frac{N(N-2)}{4}\Big(\frac{2\lambda_p}{1-|y\lambda_p+x_p|^2}\Big)^2v_p\to 0,
\quad\text{as } p\to  2^{*},
\\
\Big(\frac{1-|\lambda_py+x_p|^2}{2}\Big)^{\frac{(N-2)p-2N}{2}}\to 1,
\quad\text{as }p\to  2^{*}.
\end{gather*}
Therefore, given any radius $l$, we have
$B(0,2l)\subset B(0,\frac{d}{\lambda_p})$ for $p$
close to ~$2^{*}$. By the $L^{r}$-estimates in the theory of elliptic
equation (see \cite{GT}, for example), we can find uniform bounds for
$\|w_p\|_{W^{2,r}(B(0,2l))}(r>n)$. 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
\begin{equation}\label{4.9}
-\Delta w=(\ln\frac{1+|x_{0}|}{1-|x_{0}|})^{\alpha}L(2^{*})w^{2^{*}-1}.
\end{equation}
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
\begin{equation}\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<w\leq1,\quad  w(0)=1
\end{gathered}
\end{equation}
where $0<c=(\ln\frac{1+|x_{0}|}{1-|x_{0}|})^{\alpha}L(2^{*})<1$,
 since $0<|x_{0}|<\frac{e-1}{e+1}$. Let $z=c^{\frac{1}{2^{*}-2}}w$, then
\begin{equation}\label{4.11}
\begin{gathered}
-\Delta z=z^{2^{*}-1},\quad y\in R^N\\
z\to 0\quad\text{as }|y|\to \infty\\
0<z\leq c^{\frac{1}{2^{*}-2}},\quad z(0)=c^{\frac{1}{2^{*}-2}}.
\end{gathered}
\end{equation}
Hence $z(y)=\varepsilon^{\frac{2-N}{2}}U(\frac{x}{\varepsilon})$, where
$\varepsilon$ is determined by $c$.

By Corollary \ref{cor:3.1} and Fatou's lemma,  we have
\begin{equation} \label{4.7}
\begin{split}
S^{N/2}
&=\int_{R^N}|\nabla z|^2dx=c^{\frac{2}{2^{*}-2}}\int_{R^N}|\nabla w|^2dx\\
&\leq c^{\frac{2}{2^{*}-2}}\lim_{p\to 2^{*}}\int_{\Omega'_p}|\nabla w_p|^2dx\\
&= c^{\frac{2}{2^{*}-2}}\lim_{p\to 2^{*}}\int_{\Omega'}|\nabla v_p|^2dx\\
&=c^{\frac{2}{2^{*}-2}}S^{N/2}<S^{N/2}
\quad\text{as } p\to 2^{*},
\end{split}
\end{equation}
which is impossible.
Thus case (1) cannot occur and $x_{0}$ must be on $\partial\Omega'$.
Now we straighten $\partial\Omega'$ in a neighorhood of $x_{0}$
by a non-singular $C^{1}$ change of coordinates as in \cite{GS}:

 Let $x_{n}=\psi(x')$ $(x'=(x_{1},\dots ,x_{N-1}))$.
$\psi\in C^{1}$ be the equation of $\partial\Omega'$. Define a new coordinate
system
\begin{equation}\label{4.13}
y_{i}=x_{i}~(i=1,\dots ,N-1),\quad y_{N}=x_{N}-\psi(x')
\end{equation}
Then $v_p$ is again a solution of an equation of type (1),
and $\partial\Omega'$ is contained in the hyperplane $x_{N}=0$.
Let $d_p$ be the distance from $x_p$ to $\partial\Omega'$
(i.e. $d_p=x_p\cdot e_{N}$). Note that for $p$ close to $2^{*}$,
$w_p$ is well-defined in
$B(0,\frac{\delta}{\lambda_p})\cap\{y_{n}>-\frac{d_p}{\lambda_p}\}$
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
\begin{equation} \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},\\
0<w\leq1,\quad w(0)=1,\quad y\in R_{s}^N.
\end{gathered}
\end{equation}
By a translation, noting the fact that equation
\begin{equation}
\begin{gathered}
-\Delta w =cw^{2^{*}-1},\quad
y\in R^N_{+}=\{y=(y_{1},\dots ,y_{N-1},y_{N})|y_{N}>0)\},\\
w(y)=0,\quad y\in \partial R^N_{+}
\end{gathered}
\end{equation}
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
\begin{equation} \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,\\
0<w\leq1,\quad w(0)=1.
\end{gathered}
\end{equation}
If $L(2^{*})=0$ or $L(2^{*})=\beta$, $0<\beta<1$, just as done in case (1)
we get the contradiction $w\equiv0$ or \eqref{4.7} respectively.
So $L(2^{*})=1$, which implies that $w$ solves the equation
\begin{equation}\label{4.15}
\begin{gathered}
-\Delta w=w^{2^{*}-1},\quad y\in R^N\\
w(y)\to  0,\quad |y|\to \infty\\
0<w\leq1,\quad w(0)=1.
\end{gathered}
\end{equation}
Hence $w=\varepsilon^{\frac{2-N}{2}}U(\frac{y-y_{0}}{\varepsilon})$
for some $\varepsilon>0$, $y_{0}\in R^N$.
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
\begin{equation}\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}
\end{equation}
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 $p\to 2^{*}$,
\begin{equation} \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}
\end{equation}
The last equality follows from the weak convergence of $w_p$
in $H^{1}(\Sigma)$ and the decay of $U$ at infinity.

 As $p\to 2^{*}$,
\begin{equation} \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)
\end{equation}
 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
\begin{equation}\label{4.17}
J(v_p)=\frac{1}{N}S^{N/2}+o_{(2^{*}-p)}(1)\ as \ p\to 2^{*}.
\end{equation}
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
\begin{equation}\label{4.18}
\lim_{p\to 2^{*}}\int_{\Omega'}|\nabla(v_p-U_{\lambda_p,x_p})|^2=0.
\end{equation}
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
\begin{equation}\label{4.19}
\frac{|x_p^{1}-x_p^2|}{\lambda_p}\to +\infty
\end{equation}
or
\begin{equation}\label{4.20}
\frac{|x_p^{1}-x_p^2|}{\lambda_p}\leq c<+\infty
\end{equation}
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
\begin{equation}\label{4.21}
\lim_{p\to 2^{*}}\int_{\Omega'}|\nabla(U_{\lambda_p, x_p^{1}}
-U_{\lambda_p,x_p^2})|^2=0.
\end{equation}
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
\begin{equation}
0=2S^{N/2}-2\lim_{p\to 2^{*}}\int_{(\Omega')_p^{1}}
\nabla U\nabla U_{1,z_p}.
\end{equation}
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 $p\to 2^{*}$.
Thus part (i) of Theorem \ref{thm:1.3} is proved.
$\qed$

From Theorem \ref{thm:1.3},  we can obtain easily the following result.

\begin{corollary}\label{cor:4.1}
For $p$ close to $2^{*}$, the ground state  solution of \eqref{eq:1.1}
is not radially symmetric.
\end{corollary}

\subsection*{Acknowledgements}
This work was supported  by National Natural Sciences Foundation
of China (No. 11201140) and by Program for excellent talents in
Hunan Normal University (No. ET12101).

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