\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{equation}
\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}
\end{equation}
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 $0\leq q<p$.
The purpose of the present paper is to extend (partially) the results
obtained by \cite{J} to the case where $f$ can be singular.

This paper is organized as follows. In section 2, we recall some
preliminaries results. In section 3, we give the proof of our theorem using
mountain pass Theorem.

\section{Notation and Preliminaries}

We make use the following notation:

$L^{p}(\Omega) $, $1\leq p\leq \infty$, denote Lebesgue
spaces, the norm  $L^{p}$ is denoted by
 $\Vert\cdot\Vert _{p}$  for $1\leq p\leq \infty $;

$D^{1,2}( \mathbb{R}^{N}) $ denotes the closure space of
$C_{0}^{\infty }( \mathbb{R}^{N}) $ with respect the norm 
$\Vert \cdot\Vert _{D^{1,2}( \mathbb{R}^{N}) }:=\big(
\int_{\mathbb{R}^{N}}|\nabla u|^{2}dx\big) ^{1/2}$;

$B_{r}( 0) $ is the ball centred at $0$ with radius $r$;

$C$, $C_{1}$, $C_{2}$ will denote various positive constants;

On $H_{0}^{1}( \Omega) $ we use the norm 
\begin{equation*}
\Vert u\Vert _{\mu }=\Big( \int_{\Omega }( |\nabla
u|^{2}-\mu \frac{u^{2}}{|x|^{2}}) dx\Big) ^{1/2}.
\end{equation*}
By Hardy's inequality \cite{H}, this norm is equivalent to the usual norm 
of $H_{0}^{1}( \Omega) $.
Let 
\begin{equation*}
\mathcal{F}=\big\{f:\Omega \to \mathbb{R}^{+}: \lim_{|x|\to
0}|x|^{2}f(x)=0\quad \text{with }f\in L_{\mathrm{loc}}^{\infty }(\Omega
\backslash \{0\})\big\};
\end{equation*}
for $0\leq \beta <2$, we set 
\begin{equation*}
\mathcal{F}_{2,\beta }=\big\{f\in \mathcal{F}:0
<\lim_{|x|\to 0}|x|^{\beta }f(x)<\infty \big\}\,.
\end{equation*}
Now, we recall the following results.

\begin{lemma}[\cite{CR}] \label{lem1}
 Let $0\leq \mu <\overline{\mu }=( \frac{N-2}{2})^{2}$,
$\lambda \in \mathbb{R}^{+}$, $f\in \mathcal{F}$.
Then the eigenvalue problem
\begin{gather*}
-\Delta u-\mu \frac{u}{|x|^{2}}=\lambda f( x)u
\quad \text{in }\Omega \\
u=0 \quad \text{on }\partial \Omega%
\end{gather*}
admits a nontrivial weak solutions in $H_{0}^{1}( \Omega )$
corresponding to
$\lambda \in ( \lambda _{\mu }^{k}( f))_{k=1}^{\infty }$ where
$0<\lambda _{\mu }^{1}( f)<\lambda _{\mu }^{2}( f)\leq \lambda _{\mu }^{3}( f)
\leq \dots \to +\infty $.
\end{lemma}

\begin{lemma}[\cite{CR}] \label{lem2}
 Let $\Omega $ be a bounded domain in $\mathbb{R}^{N}$ and
$f\in \mathcal{F}$. Then the embedding $H_{0}^{1}( \Omega )
\hookrightarrow L^{2}( \Omega ,f\text{ }dx)$ is compact.
\end{lemma}

\begin{lemma}[\cite{CR}] \label{lem3}
 Let $2_{\beta }^{\ast }=\frac{2( N-\beta )}{N-2}$, if
$f\in \mathcal{F}_{2,\beta }$, $0\leq \beta <2$; then the embedding
$H_{0}^{1}( \Omega )\hookrightarrow L^{q}( \Omega ,f dx)$ is
(i) continuous for all $2\leq q\leq 2_{\beta }^{\ast }$,
(ii) compact for $2\leq q<2_{\beta }^{\ast }$.
\end{lemma}

Now, we give some examples of function $f\in \mathcal{F}$ having lower order
singularity than $|x|^{-2}$ at the origin:

\begin{itemize}
\item[(a)] Any bounded function.

\item[(b)] In a small neighbourhood of $0$, $f$ is $|x|^{-\beta }$ for 
$0<\beta <2$.

\item[(c)] $f( x)=|x|^{-\beta }/|\log |x||$ in a small neighbourhood of $0$.
\end{itemize}

\begin{definition} \label{def1} \rm
Let $c\in \mathbb{R}$, $E$ be a Banach space and
$I\in C^{1}( E,\mathbb{R})$. We say that $I$ satisfies the
Palais-Smale condition at the
level $c$, for short $(PS)_{c}$, if every sequence
$(u_{n})_{n}$ in $E$ such that $I( u_{n})\to c$
and $I'( u_{n})\to 0$ as $n\to +\infty$ in $E'$ (dual of $E$),
has a convergent subsequence in $E$.
\end{definition}

\begin{definition} \label{def2} \rm
A function $u$ in $H_{0}^{1}( \Omega )$ is said to be a weak solution of
\eqref{Plm} if $u$ satisfies
\begin{equation*}
\int_{\Omega }\Big( \nabla u\nabla v-\mu \frac{uv}{|x|
^{2}}-\lambda f( x)uvdx-u^{2^{\ast -1}}v\Big)dx=0\quad
\text{for all }v\in H_{0}^{1}( \Omega ).
\end{equation*}
\end{definition}

It is well known that the nontrivial solutions of \eqref{Plm} are equivalent to
the non zero critical points of the energy functional 
\begin{equation*}
J_{\lambda ,\mu }(u)=\frac{1}{2}\int_{\Omega }|\nabla u|^{2}dx-\frac{\mu }{2}
\int_{\Omega }\frac{u^{2}}{|x|^{2}}dx-\frac{\lambda }{2}\int_{\Omega
}f(x)u^{2}dx-\frac{1}{2^{\ast }}\int_{\Omega }|u|^{2^{\ast }}dx.
\end{equation*}
Define the constant 
\begin{equation*}
S_{\mu }=\inf_{u\in D^{1,2}\left( \mathbb{R}^{N}\right) \backslash \{0\}}
\frac{\int_{\mathbb{R}^{N}}|\nabla u|^{2}dx-\mu \int_{\mathbb{R}^{N}}
\frac{u^{2}}{|x|^{2}}dx}{\big(\int_{\mathbb{R}^{N}}|u|^{2^{\ast }}
\big)^{2/2^{\ast}}}.
\end{equation*}
It is known that $S_{\mu }$ is achieved by the family of functions%
\begin{equation*}
u_{\varepsilon }^{\ast }=\frac{C_{\varepsilon }}{(\varepsilon |x|^{\sigma
'/\sqrt{\overline{\mu }}}+|x|^{\sigma /\sqrt{\overline{\mu }}})
^{\sqrt{\overline{\mu }}}}
\end{equation*}
where $C_{\varepsilon }=(4\varepsilon N(\overline{\mu }-\mu )/(N-2))
^{\frac{\sqrt{\overline{\mu }}}{2}}$, 
$\sigma =\sqrt{\overline{\mu }}+\sqrt{\overline{\mu }-\mu }$ and 
$\sigma '=\sqrt{\overline{\mu }}-\sqrt{\overline{\mu }-\mu }$,
 see \cite{T} for the details.

Note that $u_{\varepsilon }^{\ast }$ satisfies 
\begin{equation*}
-\Delta u-\mu \frac{u}{|x|^{2}}=|u|^{2^{\ast }-2}u\quad \text{for } 
u\in D^{1,2}( \mathbb{R}^{N}) \backslash \{0\}.
\end{equation*}
Hence, we have 
\begin{equation*}
\Vert u_{\varepsilon }^{\ast }\Vert _{\mu }^{2}=\Vert u_{\varepsilon }^{\ast
}\Vert _{2^{\ast }}^{2^{\ast }}=(S_{\mu })^{N/2}.
\end{equation*}
Let $0\leq \phi (x)\leq 1$ be a function in $C_{0}^{\infty }(\Omega )$
defined as 
\begin{equation*}
\phi (x)=\begin{cases}
1 & \text{if }|x|\leq R \\ 
0 & \text{if }|x|\geq 2R,
\end{cases}
\end{equation*}
where $B_{2R}\left( 0\right) \subset \Omega $. Set 
\begin{equation}
u_{\varepsilon }=\phi (x)u_{\varepsilon }^{\ast }\quad 
\text{and}\quad v_{\varepsilon}=\frac{u_{\varepsilon }}{\Vert u_{\varepsilon }
\Vert _{2^{\ast }}},
\label{a1}
\end{equation}
so that $\Vert v_{\varepsilon }\Vert _{2^{\ast }}^{2^{\ast }}=1$.

In the present paper we prove the following result.

\begin{theorem} \label{thm1}
Let $f\in \mathcal{F}_{2,\beta }$ and $0\leq \beta <2$.
If $0\leq \mu \leq \overline{\mu }-( \frac{2-\beta }{2})^{2}$ and
$0<\lambda <\lambda _{\mu }^{1}( f)$, then \eqref{Plm}
has at least one positive solution.
\end{theorem}

\section{Proof of the main theorem}

First, we establish some lemmas.

\begin{lemma} \label{lem4}
Assume that $f\in \mathcal{F}_{2,\beta }$ and
$0<\lambda <\lambda _{\mu}^{1}( f)$. Then
$J_{\lambda ,\mu }$ satisfies $(PS)_{c}$ for all
$c<( S_{\mu })^{N/2}/N$.
\end{lemma}

\begin{proof}
Let $(u_{n})_{n}$ be a sequence such that
\begin{equation}
J_{\lambda ,\mu }(u_{n})\to c\quad \text{and}\quad J_{\lambda ,\mu
}'(u_{n})\to 0\quad \text{in }[H_{0}^{1}(\Omega) ]'\text{ as }n\to +\infty .  \label{e1}
\end{equation}
We remark that 
\begin{equation}
2J_{\lambda ,\mu }(u_{n})-\left\langle J_{\lambda ,\mu }'(u_{n}),u_{n}\right\rangle 
=(1-\frac{2}{2^{\ast }})\Vert u_{n}\Vert
_{2^{\ast }}^{2^{\ast }}\leq 2c+o(1),  \label{e2}
\end{equation}
combining \eqref{e1} and \eqref{e2} we show that $(u_{n})$ is bounded in 
$H_{0}^{1}( \Omega ) $.

From Lemmas \ref{lem2} and \ref{lem3}, and the reflexivity of 
$H_{0}^{1}(\Omega) $ we extract a subsequence, still denoted 
$u_{n}$ such that 
\begin{equation}
\begin{gathered} 
u_{n} \to u \quad\text{weakly in }H_{0}^{1}( \Omega )\\
u_{n} \to u\quad \text{in }L^{r}( \Omega )\text{ if } 1<r<2^{\ast }, \\
u_{n} \to u\quad \text{almost everywhere}, \\ 
\frac{u_{n}}{x} \to
\frac{u}{x}\quad \text{weakly in }L^{2}(\Omega ),\\ u_{n} \to u\quad
\text{strongly in }L^{2}( \Omega ,fdx). 
\end{gathered}  \label{e3}
\end{equation}
 From \eqref{e3} we deduce that 
\begin{equation}
\langle J_{\lambda ,\mu }'(u),\varphi \rangle =0\quad \text{ for all }
\varphi \in H_{0}^{1}(\Omega) ,  \label{e4}
\end{equation}
hence $u$ is a solution of \eqref{Plm}.

Denote $v_{n}:=u_{n}-u$, then the Brezis-Lieb lemma \cite{BL} implies 
\begin{equation}
\begin{gathered} \| \nabla u_{n}\| _{2}^{2} =\| \nabla u\| _{2}^{2}+\|
\nabla v_{n}\| _{2}^{2}+o( 1); \\ \| u_{n}\| _{2^{\ast }}^{2^{\ast }} =\|
u\| _{2^{\ast }}^{2^{\ast }}+\| v_{n}\| _{2^{\ast }}^{2^{\ast }}+o ( 1); \\
\int_{\Omega }\frac{u_{n}^{2}}{|x|^{2}}dx =\int_{\Omega
}\frac{u^{2}}{|x|^{2}}dx+\int_{\Omega } \frac{v_{n}^{2}}{|x|^{2}}dx+o ( 1).
\end{gathered}  \label{e5}
\end{equation}
Using \eqref{e1}, \eqref{e5} and lemma \ref{lem2}, we obtain 
\begin{equation}
J_{\lambda ,\mu }(u)+\frac{1}{2}\Vert v_{n}\Vert _{\mu }^{2}-\frac{1}{%
2^{\ast }}\Vert v_{n}\Vert _{2^{\ast }}^{2^{\ast }}=c+o(1),  \label{e6}
\end{equation}
and 
\begin{equation*}
\Vert u\Vert _{\mu }^{2}=\Vert u\Vert _{2^{\ast }}^{2^{\ast }}+\lambda
\int_{\Omega }f(x)u^{2}dx-\Vert v_{n}\Vert _{\mu }^{2}+\Vert v_{n}\Vert
_{2^{\ast }}^{2^{\ast }}+o(1).
\end{equation*}
 From \eqref{e4} it follows that 
\begin{equation*}
\Vert v_{n}\Vert _{\mu }^{2}-\Vert v_{n}\Vert _{2^{\ast }}^{2^{\ast }}=o(1).
\end{equation*}
We may therefore assume that 
\begin{equation*}
\Vert v_{n}\Vert _{\mu }^{2}\to a\quad \text{and}\quad \Vert
v_{n}\Vert _{2^{\ast }}^{2^{\ast }}\to a,
\end{equation*}
by the definition of $S_{\mu }$, we have 
\begin{equation*}
S_{\mu }\Vert v_{n}\Vert _{2^{\ast }}^{2}\leq \Vert v_{n}\Vert _{\mu }^{2},
\end{equation*}
in the limit we have 
\begin{equation*}
S_{\mu }a^{2/2^{\ast }}\leq a,
\end{equation*}
it follows that either $a=0$ or $a\geq (S_{\mu })^{N/2}$.

If $a\geq (S_{\mu })^{N/2}$ passing in the limit in \eqref{e6} we obtain 
\begin{equation*}
J_{\lambda ,\mu }(u)+\frac{1}{N}a=c
\end{equation*}
using the assumption $c<\frac{1}{N}(S_{\mu })^{N/2}$, we find 
\begin{equation}
J_{\lambda ,\mu }(u)<0.  \label{e7}
\end{equation}
On the other hand, from \eqref{e4} we obtain 
\begin{equation*}
J_{\lambda ,\mu }(u)=\frac{1}{N}\Vert u\Vert _{2^{\ast }}^{2^{\ast }}\geq 0,
\end{equation*}
which is a contradiction with \eqref{e7}. Then $u_{n}\to u$ strongly
in $H_{0}^{1}(\Omega) $.
\end{proof}

\begin{lemma} \label{lem5}
Assume that $f\in \mathcal{F}_{2,\beta }$ then
1/ There exist $\alpha ,$ $\delta >0$ 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 $J_{\lambda ,\mu }(u)=c>0$.
Since $J_{\lambda ,\mu }(u)=J_{\lambda ,\mu }(|u|)=c$, thus problem 
\eqref{Plm} admits a positive solution.
\end{proof}

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\end{document}