\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 61, 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  (login: ftp)}
\thanks{\copyright 2008 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2008/61\hfil Three solutions]
{Three solutions for singular $p$-Laplacian type equations}

\author[Z. Yang, D. Geng, H. Yan\hfil EJDE-2008/61\hfilneg]
{Zhou Yang, Di Geng, Huiwen Yan}  % keep this order, not by alphabetical order

\address{Zhou Yang \newline
 School of Math. Sci., South China Normal University,
Guangzhou 510631, China}
\email{yangzhou@scnu.edu.cn}

\address{Di Geng \newline
 School of Math. Sci., South China Normal University,
Guangzhou 510631, China}
\email{gengdi@scnu.edu.cn}

\address{Huiwen Yan \newline
 School of Math. Sci., South China Normal University,
Guangzhou 510631,  China}
\email{hwyan10@yahoo.com.cn}

\thanks{Submitted January 14, 2008. Published April 22, 2008.}
\thanks{Supported by grants 10671075 from the National Natural
 Science Foundation of China, \hfill\break\indent
 5005930 from the National Natural Science  Foundation of Guangdong, and
 \hfill\break\indent   20060574002 from the University Special
Research Fund for Ph. D. Program.}
\subjclass[2000]{35J60}
\keywords{$p$-Laplacian operator; singularity; multiple solutions}

\begin{abstract}
 In this paper, we consider the  singular $p$-Laplacian type equation
 \begin{gather*}
  -\mathop{\rm div}(|x|^{-\beta} a(x,\nabla u))
 =\lambda f(x,u),\quad \mbox{in }\Omega,\\
 u=0,\quad \mbox{on }\partial\Omega,
 \end{gather*}
 where $0\leq\beta<N-p$, $\Omega$ is a smooth bounded domain in
 $\mathbb{R}^N$  containing the origin, $f$ satisfies some growth
 and singularity conditions. Under some mild assumptions on $a$,
 applying the three critical points theorem developed
 by Bonanno, we establish the existence of at least three distinct
 weak solutions to the above  problem if $f$ admits some
 hypotheses on the behavior at $u=0$ or  perturbation property.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}

\section{Introduction}

The three critical points theorem established by Ricceri
\cite{Ricceri1} and extended by Bonanno \cite{Bonanno} has been used
by several author in the study of nonlinear boundary-value problems;
see for example \cite{B,Bonanno,Cordaro,Alexandru,Ricceri2,Yang}. In
particular, Krist\'aly, Lisei and Vargaetc \cite{Alexandru} employed
Bonanno's theorem to study the $p$-Laplacian type equation
 \begin{equation} \label{eq1.1}
\begin{gathered}
-\mathop{\rm div}(a(x,\nabla u))=\lambda f(u),\quad \mbox{in }\Omega,\\
 u=0,\quad \mbox{on }\partial\Omega,
 \end{gathered}
 \end{equation}
 where $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$ and
 $a:\Omega\times\mathbb{R}^N\to\mathbb{R}^N$ satisfies some structural
 conditions. The simplest case of this problem occurs when
$a(x,\xi)=|\xi|^{p-2}\xi$, $p>1$. In this case \eqref{eq1.1} reduces
to an equation  involving the $p$-Laplacian operator.
Under the assumptions that the nonlinear term
$f(u):\mathbb{R}\to\mathbb{R}$ is continuous,
$(p-1)$-sublinear at infinity and
$(p-1)$-superlinear at the origin,
Krist\'aly  applied  Bonanno's variational principle to \eqref{eq1.1}
and obtain the  existence of three weak solutions.

In the present paper, we investigate the existence and multiplicity of
solutions to the singular $p$-Laplacian type equation
\begin{equation} \label{eq1.2}
 \begin{gathered}
 -\mathop{\rm div}(|x|^{-\beta} a(x,\nabla u))
 =\lambda f(x,u),\quad \mbox{in }\Omega,\\
 u=0,\quad \mbox{on }\partial\Omega,
 \end{gathered}
\end{equation}
 where $0\leq\beta<N-p$, $1<p<N$ and $\Omega$ is a smooth bounded domain in
 $\mathbb{R}^N$ containing the origin.

In this paper, we use the following notation:
\begin{equation} \label{betastar}
\beta^*_1:={N\beta\over N-p},\quad
\beta^*_2:=p+\beta,\quad
\beta^*_3:=N-{N-p-\beta\over p},\quad
p^*(\beta,\alpha):={(N-\alpha)p\over N-\beta-p}.
\end{equation}
 Suppose that the potential $a:\Omega\times \mathbb{R}^N\to \mathbb{R}^N$
 satisfies the assumptions:

 Let $A=A(x,\xi):\Omega\times\mathbb{R}^N\to {\mathbb{R}}$ be a
 Carath\'eodory function, i.e.,
 measurable in $x$ and continuous in $\xi$, a.e. $x\in\Omega$;
$A(x,\xi)$ is of continuous derivative with respect to $\xi$ with
$a=\nabla_\xi A$  and satisfies the follows conditions:
\begin{itemize}
\item[(A1)] $A(x,0)=0$ a.e. $x\in\Omega$;

\item[(A2)] there are  $p>1$ and a positive constant $a_1$ such that
$$
|a(x,\xi)|\leq a_1(1+|\xi|^{p-1})\quad\mbox{for a.e. }x\in\Omega
\mbox{ and all } \xi\in \mathbb{R}^N;
$$
\item[(A3)] $A(x,\xi)$ is strictly convex in $\xi$, that is,
for $\xi,\eta\in\mathbb{R}^N$ with $\xi\neq\eta$
$$
2A\left(x,{\xi+\eta\over2}\right)<A(x,\xi)+A(x,\eta)\quad \mbox{ for
a.e. }x\in\Omega;
$$
\item[(A4)] $A(x,\xi)$ satisfies the ellipticity condition:
There exists a positive  constant $a_2$ such that
$$
A(x,\xi)\geq a_2|\xi|^p,\quad\mbox{for a.e. }x\in\Omega\quad\mbox{and all }\xi\in \mathbb{R}^N.
$$
 \end{itemize}
We suppose the singular nonlinear term $f(x,u)$ fulfils the
 following hypothesis:
 Let $f=f(x,u):\mathbb{R}^N\times {\mathbb{R}}\to {\mathbb{R}}$
 be a Carath\'eodory function and
 \begin{itemize}
 \item[(B1)] $f(x,u)$ is subcritical and $(p-1)$-sublinear at
 infinity, i.e.,
 $$
 \lim_{u\to\infty}\sup_{x\in\Omega}
 |f(x,u)||u|^{1-p}|x|^{\beta_2^*}=0.
 $$

 \item[(B2)]  There exist some $\alpha$ with
$\beta^*_1\leq \alpha<\beta^*_2$
 and a positive continuous  function $\mathcal{F}(u)$ with
$\mathcal{F}(u)(1+|u|^p)^{-1}\in L^\infty(\mathbb{R})$
 such that
 $$
  |F(x,u)|\leq \mathcal{F}(u)|x|^{-\alpha}\;\;
 \mbox{for a.e.}\;\;(x,u)\in \Omega\times \mathbb{R}.
 $$
 \end{itemize}

In the sequel we consider the weighted space
$X=\mathcal{D}^{1,p}(\Omega, |x|^{-\beta}dx)$,
which is the completion of $C^\infty_0(\Omega)$ under the norm
$(\int_\Omega|\nabla u|^p|x|^{-\beta}dx)^{1/p}$.
On $X$, we define the two functionals
\begin{equation} \label{eq1.3}
\Phi(u)=\int_\Omega A(x,\nabla u)|x|^{-\beta}dx,\quad
\Psi(u)=\int_\Omega F(x,u)dx,
\end{equation}
where $F(x,u)=\int_0^u f(x,t)dt$.

It is not difficult to see that solutions of the problem
\eqref{eq1.2} are the critical points of the variational functional
$I(u)=\Phi(u)-\lambda\Psi(u)$.
Moreover, $I(u)$ is continuous differentiable  on the space $X$,
and Fr\'echet derivation of $I(u)$ can be represented as
 \begin{equation} \label{eq1.4}
 \langle I'(u),v\rangle=
 \int_\Omega |x|^{-\beta} a(x,\nabla u)\cdot\nabla v dx-
 \lambda\int_\Omega f(x,u)v,\quad \forall\;v\in X.
 \end{equation}

According to the structural conditions of $a(x,\xi)$ and $f(x,u)$,
it is clear that the problem \eqref{eq1.2} is more general than
\eqref{eq1.1} since there exists singularity not only in nonlinear
term $f(x,u)$, but also in diverge term
$\mathop{\rm div}(|x|^{-\beta} a(x,\nabla u))$, which issues some difficulty.
We need some generalized Hardy-Sobolev imbedding result
(see Lemma \ref{lem2.1} below) in proving the P.-S. condition. Since we drop the
assumption (Ha) in \cite{Alexandru} and replace the usual
$p$-uniform convexity of $A(x,\xi)$ by strict convexity,
to show that $I(u)$ is weakly lower semicontinuous on $X$
(Lemma \ref{lem2.5}),
we have to give some subtle estimates about
the variational functional $I(u)$.

In this paper, when $f(x,u)$ is $(p-1)$-superlinear
 at the origin, the first main result we establish is:

\begin{theorem} \label{thm1.1}
Assume {\rm (A1)--(A4), (B1)--(B2)} are satisfied. Let $E=B(x_0,r)$
be a ball contained in $\Omega$, such that for some $K\not=0$,
\begin{equation} \label{C1}
\inf_{x\in E}F(x,K)>0.
\end{equation}
If $F(x,u)$ admits the asymptotic property at the origin:
\begin{equation} \label{C2}
\mathcal{F}(u)|u|^{-p}\to0\quad\mbox{as }u\to0,
\end{equation}
then, there exists an open interval $\Lambda\subset[0,+\infty)$
 and a number $R>0$ such that for every $\lambda\in \Lambda$,  equation
 \eqref{eq1.2} has at least three distinct solutions in $X$, whose $X$-norms
 are less than $R$.
\end{theorem}

Note that when $\beta=\alpha=0$ and $f(x,u)=f(u)$,
Theorem \ref{thm1.1} implies the conclusion in \cite[Theorem 2.1]{Alexandru}.

 The conclusion in Theorem \ref{thm1.1} still holds if the asymptotic
 property of $f(x,u)$ at the origin is replaced by some other properties.
To state the next result, we introduce the following notation:
\begin{equation}  \label{C3}
 c_2(s)=\inf_{x\in B(x_0,r/2)} {F(x,s)\over1+|s|^p},\quad
 c_3(s)=\sup_{|u|\geq s} \mathcal{F}(u)|u|^{-p},\quad
 c_4(s)=\sup_{|u|\leq s}\mathcal{F}(u),
\end{equation}
where $B(x_0,r)\subset\Omega$ and $s\geq0$.

\begin{theorem}  \label{thm1.2}
Assume {\rm (A1)--(A4), (B1)--(B2)} are satisfied. Let $E=B(x_0,r)$
be a ball contained in $\Omega$, such that
\begin{equation}  \label{C3b}
F(x,u)\geq0,\quad\mbox{ for a.e. }x\in E\mbox{ and all }u\in I,
\end{equation}
where $I$ is either $\mathbb{R}^+$ or $\mathbb{R}^-$.
If there exist $L>0$ and  $K\in I$  such that
\begin{equation} \label{C4}
 c_2(K)|K|^p\geq C c_4(L),\quad
 c_2(K)>C(c_3(L))^{p\over q}(c_4(L))^{q-p\over q}L^{p(p-q)\over q},
\end{equation}
 where $q=p^*(\beta,\alpha)$ and $C$ is a certain positive constant
only dependent on  $p$, $\beta$, $\alpha$, $N$, $E$, $a_1$ and $a_2$.
Then the conclusion in Theorem \ref{thm1.1} remains valid.
\end{theorem}

\begin{remark} \label{rmk1.3}\rm
 The above result is new even in the case of $\beta=\alpha=0$.
Moreover, by the method similar to \cite{Yang}, we
 can show a more general result.
\end{remark}

\begin{remark} \label{rmk1.4} \rm
If we fix some $L$ and keep $c_2(K)/c_3(L)$ less
 than a fixed constant, then assumption \eqref{C4} holds when
 $K>L$ and $c_2(K)$ is large enough.
\end{remark}

\section{Preliminaries}

 Firstly, we recall the generalized Hardy-Sobolev imbedding
 theorem, which can be deduced from Caffarelli-Kohn-Nirenberg
 inequality (see \cite{Caffarelli,Xuan}).

\begin{lemma} \label{lem2.1}
Suppose that $\beta^*_1\leq\widetilde{\alpha}\leq\beta^*_2$
and $\beta^*_1\leq\widehat{\alpha}<\beta^*_3$.
 Let $U$ be an arbitrary smooth bounded domain in  ${\mathbb{R}^N}$ containing the
 origin. We have
\begin{itemize}
\item[(i)] There exists a constant $S_{\widetilde{\alpha}}>0$,
such that for any
 $u\in\mathcal{D}^{1,p}({\mathbb{R}^N}, |x|^{-\beta}dx)$, there holds
 $$
 S_{\widetilde{\alpha}}\|u\|_{L^{p^*(\beta,\widetilde{\alpha})}
 ({\mathbb{R}^N},|x|^{-\widetilde{\alpha}}dx)}^p\leq\|u\|_
 {\mathcal{D}^{1,p}({\mathbb{R}^N},|x|^{-\beta}dx)}^p\;,
 $$
where $L^p(U,|x|^{-\alpha}dx)$ is $L^p$ space with $|x|^{-\alpha}$
as weight.

\item[(ii)] For $1\leq\widetilde q\leq p^*(\beta,\widetilde{\alpha})$,
 there exists a constant $ S_{\widetilde{q},\widetilde{\alpha}}>0$
 such that for any $u\in\mathcal{D}$$^{1,p}(U,|x|^{-\beta}dx)$,
 there holds
 $$
 S_{\widetilde{q},\widetilde{\alpha}}\|u\|_{L^{\widetilde q}
 (U,|x|^{-\widetilde{\alpha}}dx)}^p
 \leq\|u\|_{\mathcal{D}^{1,p}(U,|x|^{-\beta}dx)}^p\;,
 $$
 Moreover, $S_{\widetilde{\alpha}}=S_{\widetilde{q},\widetilde{\alpha}}$
 is independent of the domain $U$ provided
 $\widetilde{q}= p^*(\beta,\widetilde{\alpha})$.

\item[(iii)]
 $\mathcal{D}$$^{1,p}(U,|x|^{-\beta}dx)$ compactly imbeds into
 $L^{\widehat{q}}(U,|x|^{-\widehat{\alpha}})$ provided
$1\leq\widehat q<p^*(\beta,\widehat{\alpha})$.
\end{itemize}
 \end{lemma}

\begin{remark} \label{rmk2.1} \rm
\begin{itemize}
\item[(i)] The first assertion in the lemma is a special case of
 Caffarelli-Kohn-Nirenberg inequality.
 Particularly, let $\beta=0,\widetilde{\alpha}=\beta_2^*=p$, one get Hardy inequality;
 furthermore, let $\beta=\widetilde{\alpha}=\widehat{\alpha}=0$,
the lemma leads to Sobolev theorem.

\item[(ii)] There are various forms of description about the imbedding,
 such as \cite{Xuan} and references therein. We use the form because
it looks like
 a generalization of Hardy-Sobolev imbedding theorem.
\end{itemize}
\end{remark}

For the reader's convenience, we give the proof of the above lemma,
which is similar to \cite{Xuan}.

\begin{proof}[Proof of lemma  \ref{lem2.1}]
 Assertion (i) can be directly deduced from main results in
 \cite[Theorem]{Caffarelli}. In fact, choose the parameters
$n$, $p$, $\gamma=\beta$, $r=q$, $\alpha$, $a$ and $\sigma$ in
\cite{Caffarelli} as
$N$, $p$, $-\widetilde{\alpha}/p^*(\beta,\widetilde{\alpha})$,
$p^*(\beta,\widetilde{\alpha})$, $-\beta/p$, $1$ and
$-\widetilde{\alpha}/p^*(\beta,\widetilde{\alpha})$, respectively.
Then it is not difficult to verify the assumptions in \cite{Caffarelli} and
 thus (i) follows.

(ii) Recalling that $\beta^*_1\leq\widetilde{\alpha} \leq\beta^*_2$ and
 $1\leq \widetilde{q}\leq p^*(\beta,\widetilde{\alpha})$,
 we have
\[
   \int_{U}|u|^{\widetilde{q}}|x|^{-\widetilde{\alpha}}dx
   \leq \Big(\int_{U}|u|^{p^*(\beta,\widetilde{\alpha})}
   |x|^{-\widetilde{\alpha}}dx\Big)
   ^{\widetilde{q}/p^*(\beta,\widetilde{\alpha})}
   \Big(\int_{U}|x|^{-\widetilde{\alpha}}dx\Big)
   ^{(p^*(\beta,\widetilde{\alpha})-\widetilde{q})
   /p^*(\beta,\widetilde{\alpha})}.
\]
 Since $U$ is bounded and $\widetilde{\alpha}\leq \beta_2^*<\beta_3^*<N$,
the above inequality and conclusion (i) imply the required result.
Employing the scaling method, one can discover
 that the constant
 $S_{\widetilde{\alpha}}=S_{\tilde q,\widetilde{\alpha}}$
is independent of the domain $U$ if $\tilde q=p^*(\beta,\tilde\alpha)$.

(iii) First we prove that
 $\mathcal{D}$$^{1,p}(U,|x|^{-\beta}dx)$
 imbeds into $L^{\widehat{q}}(U,|x|^{-\widehat{\alpha}}dx)$.  According
 to assertion (ii), it is sufficient to demonstrate the imbedding when
 $\beta_2^*<\widehat{\alpha}<\beta_3^*$. Indeed, noting that
 $1=p^*(\beta,\beta_3^*)<\widehat{q}<p^*(\beta,\beta_2^*)=p$,
 we calculate
 $$
 \int_{U}|u|^{\widehat{q}}|x|^{-\widehat{\alpha}}dx
 \leq \Big(\int_{U}|u|^{p}|x|^{-\beta_2^*}dx\Big)^{\widehat{q}/p}
 \Big(\int_{U}|x|^{-\tau}dx\Big)
 ^{(p-\widehat{q})/p},
 $$
where $\tau=(\widehat{\alpha}-\beta_2^*\widehat{q}/p)p/(p-\widehat{q})$.
Since $\widehat{q}<p^*(\beta,\widehat{\alpha})$, we obtain
 $$
 \widehat{\alpha}<N-{N-\beta-p\over p}\widehat{q},\quad
 \tau<\Big(N-{N-\beta-p\over p}\widehat{q}-
 {\beta+p\over p}\widehat{q}\Big){p\over p-\widehat{q}}=N,
 $$
 which means that $\mathcal{D}$$^{1,p}(U,|x|^{-\beta}dx)$
 imbeds into $L^{\widehat{q}}(U,|x|^{-\widehat{\alpha}}dx)$.

It remains to prove the imbedding is compact.
Assume that the sequence $\{u_n\}_{n=1}^\infty$ is bounded in
 $\mathcal{D}^{1,p}(U,|x|^{-\beta}dx)$, it is sufficient to
 show that there exists a subsequence, still denoted by itself, such that
 $u_n$ strongly converges to $u$ in
 $L^{\widehat{q}}(U,|x|^{-\widehat{\alpha}})$ as
 $n\to\infty$.

In fact, since $U$ is bounded, we observe
\begin{align*}
 \|u\|_{\mathcal{D}^{1,p}(U)}^p
 &=\int_{U}|\nabla u|^p\,dx\\
 &\leq  (\mathop{\rm diam}U)^\beta\int_{U}|\nabla u|^p|x|^{-\beta}dx\\
 &\leq(\mathop{\rm diam}U)^\beta\|u\|_{\mathcal{D}^{1,p}(U,|x|^{-\beta}dx)}^p.
\end{align*}
 So, $\{u_n\}_{n=1}^\infty$ is also bounded in
 $\mathcal{D}$$^{1,p}(U)$ and there exists a subsequence,
 still denoted by itself, weakly converging to some $u$ in
 $\mathcal{D}$$^{1,p}(U)$. Remembering that
$1< \widehat{q}< p^*(\beta,\widehat{\alpha}) \leq p^*(\beta,\beta^*_1)
 =Np/(N-p)$, we conclude that $u_n$ strongly converges to $u$ in
 $L^{\widehat{q}}(U)$ from the Sobolev theorem.

Choose a sequence of positive numbers $\{\rho_m\}$ such that
 $\rho_m\to 0$ as $m\to\infty$ and
 $\overline{B}_{\rho_m}(0)\subset U$ for all $m\in{\mathbb Z}^+$.
Then  we deduce
 $$
 \int_{U\setminus\overline{B}_{\rho_m}(0)}
 |u_n-u|^{\widehat{q}}|x|^{-\widehat{\alpha}}dx\leq
 \rho_m^{-\hat\alpha} \|u_n-u\|^{\widehat{q}}_{L^{\widehat{q}}
 (U\setminus\overline{B}_{\rho_m}(0))}
 \leq C_m\|u_n-u\|^{\widehat{q}}_{L^{\widehat{q}}(U)}.
 $$
 On the other hand, recalling $\widehat{\alpha}<N$, we compute
 $$
 \int_{\overline{B}_{\rho_m}(0)}
 |u_n-u|^{\widehat{q}}|x|^{-\widehat{\alpha}}dx\leq
 \|u_n-u\|_{L^\tau(U,|x|^{-\widehat{\alpha}}dx)}
 ^{\widehat{q}}\Big(\int_{\overline{B}_{\rho_m}(0)}|x|
 ^{-\widehat{\alpha}}dx\Big)^{(\tau-\widehat{q})/\tau},
 $$
 here $\tau=(\widehat{q}+p^*(\beta,\widehat{\alpha}))/2>\widehat{q}$.
 Combining the above two inequalities, we obtain
 $$
0\leq\int_U|u_n-u|^{\widehat{q}}|x|^{-\widehat{\alpha}}dx
\leq C_m\|u_n-u\|^{\widehat{q}}_{L^{\widehat{q}}(U)}
 +C\Big(\int_{\overline{B}_{\rho_m}(0)}|x|
 ^{-\widehat{\alpha}}dx\Big)^{(\tau-\widehat{q}\;)/\tau}.
 $$
 First let $n\to\infty$, then $m\to\infty$, and we derive that $u_n$
strongly converges  to $u$ in $L^{\widehat{q}}(U,|x|^{-\widehat{\alpha}})$.
\end{proof}

Secondly, we review  Bonanno's three critical points theorem
 (see \cite{Bonanno}), which is the main variational tool in this paper.

\begin{lemma} \label{lem2.2}
Let $\mathcal{X}$ be a separable and reflexive real Banach
 space, and let $\phi,\psi:{\mathcal{X}}\to \mathbb{R}$ be two continuously
 G\^ateaux differentiable functionals. Assume that
 \begin{itemize}
 \item[(D1)] There exists a function $u_0\in{\mathcal{X}}$ such that
$\phi(u_0)=\psi(u_0)=0$
 and $\phi(u)\geq0$ for every $u\in{\mathcal{X}}$.
 \item[(D2)] There exists a function $u_1\in{\mathcal{X}}$ and a
positive number $\rho$ such that
 \begin{equation} \label{eq2.1}
 \rho<\phi(u_1),\quad
 \sup_{\phi(u)<\rho}\psi(u)<\rho {\psi(u_1)\over\phi(u_1)}.
 \end{equation}

\item[(D3)] Further, put
 $$
 \gamma=\xi\rho\Big[\rho {\psi(u_1)\over\phi(u_1)}-
 \sup_{\phi(u)<\rho}\psi(u)\Big]^{-1},
 $$
 with $\xi>1$, and suppose that for  every $\lambda\in[0,\gamma]$,
 the functional $\phi(u)-\lambda\psi(u)$
 is sequentially weakly lower semicontinuous, satisfies the P.-S. condition
 and
 \begin{equation} \label{eq2.2}
 \lim_{\|u\|\to+\infty}\Big[\phi(u)-\lambda\psi(u)\Big]=+\infty.
 \end{equation}
 \end{itemize}
Then, there exists an open interval $\Lambda\subset[0,\gamma]$
 and a number $R>0$ such that, for any $\lambda\in \Lambda$, the equation
 $\phi'(u)-\lambda\psi'(u)=0$ admits at least three solutions
in ${\mathcal{X}}$ whose  norms are less than $R$.
 \end{lemma}

 In the sequel,
 by setting ${\mathcal{X}}=X=\mathcal{D}^{1,p}(\Omega,|x|^{-\beta}dx)$,
$\phi(u)=\Phi(u)$, $\psi(u)=\Psi(u)$ and $\xi=+\infty$
 we show that the variational functional $I(u)$ satisfies
all assumptions in Lemma \ref{lem2.2}.

\begin{lemma} \label{lem2.3}
Suppose that the assumptions {\rm (B1), (B2)} are satisfied. Then
$\Psi(u)$ is weakly continuous on $X$, i.e., if $u_n$ weakly
 converges to $u$ in $X$, $\Psi(u_n)$ converges to $\Psi(u)$.
\end{lemma}

\begin{proof}
According to assumptions {\rm (B1), (B2)}, it is not difficult
 to deduce that, for each $\epsilon>0$, there exists some  positive
 number $M_\epsilon$ such that
 \begin{gather} \label{eq2.3}
 |f(x,u)u|+|F(x,u)|\leq \epsilon|u|^p|x|^{-\beta_2^*},\quad
 \mbox{a.e. }x\in \Omega \mbox{ and all }|u|\in[M_\epsilon,+\infty);\\
\label{eq2.4}
|f(x,u)u|+|F(x,u)|\leq \epsilon|u|^p|x|^{-\beta_2^*}
 +C_\epsilon|u||x|^{-\alpha},\quad
 \mbox{a.e. }x\in \Omega\mbox{ and all }u\in \mathbb{R},
 \end{gather}
 here $C_\epsilon$ is a positive number dependent only on $\epsilon$.

Assume that $u_n$  converges weakly to $u$ in $X$, then for any
 $\epsilon\geq0$, we conclude
 \begin{align*}
 |F(x,u_n)-F(x,u)|&\leq |f(x,\theta u+(1-\theta)u_n)||u_n-u|\\
 &\leq (\epsilon|u|^{p-1}|x|^{-\beta_2^*}+\epsilon|u_n|^{p-1}|x|^{-\beta_2^*}
 +\overline{C}_\epsilon|x|^{-\alpha})|u_n-u|,
 \end{align*}
 where $0<\theta<1$. The definition of $\Psi(u)$ thus implies that
 \begin{align*}
 |\Psi(u_n)-\Psi(u)|
&\leq\int_\Omega|F(x,u_n)-F(x,u)|dx \\
&\leq \int_\Omega\Big(\epsilon {|u|^{p-1}+|u_n|^{p-1}
 \over|x|^{\beta_2^*}}+ {\overline{C}_\epsilon\over
  |x|^\alpha}\Big)|u_n-u|\,dx\\
&\leq C\epsilon(\|u_n\|_X^p+\|u\|_X^p)
 +\overline{C}_\epsilon\|u_n-u\|_{L^1(\Omega;|x|^{-\alpha}dx)}.
 \end{align*}
 Since $X$ compactly imbeds into $L^1(\Omega;|x|^{-\alpha}dx)$,
 taking $n\to\infty$, we obtain
 $$
 \limsup_{n\to\infty}|\Psi(u_n)-\Psi(u)|\leq
  C\epsilon\|u\|_X^p.
 $$
 Let $\epsilon\to0^+$ in the above inequality, and
 the conclusion in the lemma follows.
\end{proof}

\begin{lemma} \label{lem2.4}
Suppose that the assumptions {\rm (A1)--(A4), (B1)--(B2)} are
satisfied. Then $I(u)$ is weakly lower semicontinuous on $X$.
\end{lemma}

\begin{proof}
Owing to previous lemma, it suffice to show
 weakly lower semicontinuity of $\Phi(u)$ on $X$.
We argue by contradiction, assume that $\{u_n\}$ is a function
 sequence weakly converging to $u$ in $X$, but there is
 a subsequence $u_{n_k}$ such that
 $\lim_{k\to\infty}\Phi(u_{n_k})>\Phi(u)$. Without
 loss of generalization, one can assume that
 $$
 \Phi(u_{n_k})>\Phi(u)+\delta,\quad\mbox{for }k=1,2,\dots,
 $$
where $\delta$ is a positive number.

In view of Mazur theorem, there exists a sequence $\{v_m\}$ strongly
 converging to $u$ in $X$, where $v_m$ is a convex combination of
 finitely many $u_{n_k}$; i.e., for any $m\in{\mathbb Z}^+$,
 $$
 v_m=\sum_{i=1}^m \alpha_{m i} u_{n_{k_i}},
 \quad\mbox{with } \alpha_{m i}>0,\;\;
 \sum_{i=1}^m \alpha_{m i}=1.
 $$
Since $A(x,\xi)$ is convex with respect to $\xi$,
  we then derive
\begin{align*}
 \Phi(v_m)&\geq \sum_{i=1}^m \alpha_{m i}
 \int_\Omega A(x,\nabla u_{n_{k_i}})
 |x|^{-\beta}dx \\
&=  \sum_{i=1}^m \alpha_{m i}
 \Phi(u_{n_{k_i}})>\Phi(u)+\delta,\quad
 \mbox{for }m=1,2,\dots,
\end{align*}
 which contradicts that $\{v_m\}$ strongly converges to $u$ in $X$.
\end{proof}

 \begin{lemma} \label{lem2.5}
Suppose that the assumptions {\rm (A1)--(A4), (B1)--(B2)}  are
satisfied. Then $I(u)$ satisfies the P.-S. condition.
\end{lemma}

\begin{proof}
Suppose that $\{u_n\}\subset X$ is a P.-S. sequence for $I(u)$,
 that is,  $\{I(u_n)\}$ is bounded, and
 $\|I'(u_n)\|_{X^*}\to 0$ as $n\to0$, where
 $X^*$ is the dual space of $X$.

 We claim that $\{u_n\}$ admits a strongly convergent subsequence.
 Firstly, we show that $\{u_n\}$ is bounded in $X$. In fact, combining
 assumption {\rm (A4)}, (\ref{eq2.4}) and Lemma \ref{lem2.1},
 we calculate
 \begin{align*}
 C\geq I(u_n)
&=\int_\Omega A(x,\nabla u_n)|x|^{-\beta}dx-
 \lambda \int_\Omega F(x, u_n)dx\\
&\geq a_2\int_\Omega |\nabla u_n|^p|x|^{-\beta}dx
 -\lambda \int_\Omega ( \epsilon|u|^p|x|^{-\beta_2^*}
 +C_\epsilon|x|^{-\alpha})dx\\
&\geq (a_2-\lambda S^{-1}_{\beta_2^*}\epsilon)\|u_n\|_X^p
 -\overline C_\epsilon.
\end{align*}
 Fix $\epsilon>0$ small enough that $a_2-\lambda
 S_{\beta_2^*}^{-1}\epsilon \geq a_2/2$, then we discover that
  $\{u_n\}$ is bounded in $X$.
There thus exists a subsequence of $\{u_n\}$, still denoted by itself,
 such that $\{u_n\}$ weakly converges to $u$ in $X$. Moreover, without
 loss of generalization, one can assume that $f(x,u_n)$ weakly converges
 to $f(x,u)$ in $X^*$.

We next demonstrate that there exists a subsequence of $\{u_n\}$,
 still denoted by itself, such that
 \begin{equation} \label{eq2.5}
 \lim_{n\to\infty}\nabla u_n=\nabla u
 \quad \mbox{a.e. in }\Omega.
 \end{equation}
Indeed, the facts that $\{u_n\}$
 is bounded in $X$ and $\|I'(u_n)\|_{X^*}\to 0$ as
 $n\to\infty$ implies that
 \begin{equation} \label{eq2.6}
 \langle I'(u_n)-I'(u), u_n-u\rangle=\langle I'(u_n), u_n-u\rangle
 -\langle I'(u), u_n-u\rangle=o(1),\quad\mbox{as }n\to\infty.
 \end{equation}
 Furthermore, repeat the argument in the proof of Lemma \ref{lem2.3},
and it is easy to deduce
\begin{equation}    \label{eq2.7}
\begin{aligned}
 J(u,u_n)&:=\int_\Omega[f(x,u_n)-f(x,u)][u_n-u]dx\\
 &= \int_\Omega f(x,u_n)(u_n-u)\, dx-
 \int_\Omega f(x,u)(u_n-u)\,dx=o(1),
 \end{aligned}
\end{equation}
as $n\to\infty$.  On the other hand,
 \begin{equation} \label{eq2.8}
 \langle I'(u_n)-I'(u), u_n-u\rangle=\int_\Omega H(x,u,u_n)|x|^{-\beta}dx
 -\lambda J(u,u_n),
 \end{equation}
 where
 $$
 H(x,u,u_n):=[ a(x,\nabla u_n)
 -a(x,\nabla u)]\cdot[\nabla u_n-\nabla u].
 $$
Combining (\ref{eq2.6}), (\ref{eq2.7}) and  (\ref{eq2.8}),
 we obtain
 \begin{equation} \label{eq2.9}
 \lim_{n\to\infty}\int_\Omega H(x,u,u_n)|x|^{-\beta}dx=0.
 \end{equation}
Notice that $H(x,u,u_n)\geq0$ since $A(x,\xi)$ is convex in $\xi$.
So, (\ref{eq2.9}) implies that there exists a subsequence of
 $\{u_n\}$, still denoted by itself, such that $H(x,u,u_n)\to0$
 a.e. in $\Omega$ as $n\to\infty$. Hence, (\ref{eq2.5}) follows
 from the strict convexity of $A(x,\xi)$.

Then, we prove that there exists a subsequence of
 $\{u_n\}$, still denoted by itself, such that
 \begin{equation} \label{eq2.10}
 \lim_{n\to\infty}
 \int_\Omega |x|^{-\beta} a(x,\nabla u_n)\cdot \nabla u_n\, dx=
 \int_\Omega |x|^{-\beta} a(x,\nabla u)\cdot \nabla u \,dx.
 \end{equation}
According to the growth condition (A2) and (\ref{eq2.5}),
 we can assume that $a(x,\nabla u_n)$ weakly converges to $a(x,\nabla u)$
 in $X^*$, maybe a subsequence of $\{u_n\}$. Recalling that $f(x,u_n)$
 weakly converges to $f(x,u)$ in $X^*$, we infer that
 $I'(u_n)$ weakly converges to $I'(u)$ in $X^*$. Hence, as
 $n\to\infty$, we deduce
 \begin{align*}
 o(1)
&=\langle I'(u_n), u_n-u\rangle-\langle I'(u_n)-I'(u),u\rangle\\
&=\langle I'(u_n), u_n\rangle-\langle I'(u),u\rangle
 \\
 &=\int_\Omega |x|^{-\beta} [ a(x,\nabla u_n)\cdot
 \nabla u_n -a(x,\nabla u)\cdot \nabla u]dx\\
 &\quad -\lambda\int_\Omega [f(x,u_n)u_n-f(x,u)u]\,dx.
 \end{align*}
 Repeating the procedure as in the proof of (\ref{eq2.7}), we can
 achieve (\ref{eq2.10}).

 On the other hand, since $A(x,\xi)$ is convex with $A(x,0)=0$ and
 satisfies elliptic condition, we observe
\[
  a(x,\xi)\cdot\xi\geq A(x,\xi)\geq a_2|\xi|^p,\quad
  \mbox{for all }\xi\in \mathbb{R}^N,
\]
 which implies  $a_2|\nabla u_n|^p$ and $a_2|\nabla u|^p$ being
dominated by
 $a(x,\nabla u_n)\cdot \nabla u_n,\;a(x,\nabla u)\cdot \nabla u$, respectively.
 Combining (\ref{eq2.5}), (\ref{eq2.10}) and the dominated convergence
theorem, we conclude that $\nabla u_n$ converges to $\nabla u$ in
 $L^p(\Omega,|x|^{-\beta}dx)$, that is
 $u_n$ strongly converges to $u$ in $X$.
\end{proof}

 \section{Proof of the main results}

To prove Theorems \ref{thm1.1} and \ref{thm1.2}, we set notation as follows:
\begin{equation} \label{eq3.1}
\Pi(F;M)=M^{p-q}\Big({\rho\over a_2 S_\alpha}\Big)^{q/p}
\sup_{|u|\geq M}\mathcal{F}(u)|u|^{-p}
+ \mu(\Omega)\sup_{|u|\leq M}\mathcal{F}(u),
\end{equation}
where $\mu(\Omega):=\int_\Omega |x|^{-\alpha}\,dx$
 and $q=p^*(\beta,\alpha)$ as defined in (\ref{betastar}).
One can establish the next result.

\begin{lemma} \label{lem3.6}
Suppose that the hypothesis {\rm (B2)} and {\rm (A4)}
are satisfied. For every $u\in X$ with
$\Phi(u)\leq\rho$, we have
$$
\Psi(u)\leq\Pi(F;M).
$$
\end{lemma}

\begin{proof}
 According to assumption (A4) and Lemma \ref{lem2.1},
 for every $u\in \Phi^{-1}(-\infty,\rho]$, we have
 \begin{equation} \label{eq3.2}
 \|u\|^p_X\leq {\Phi(u)\over a_2}\leq {\rho\over a_2}\,,\quad
 \|u\|_\alpha^q\leq {\|u\|_X^q\over S_\alpha^{q/p}}\leq
 \Big({\rho\over a_2 S_\alpha}\Big)^{q/p},
\end{equation}
where $\|u\|_\alpha^q:=\int_\Omega|u|^q|x|^{-\alpha}dx$.
By setting $\Omega_M:=\{x\in\Omega:|u(x)|\geq M\}$, we can deduce
 \begin{equation} \label{eq3.3}
 \mu(\Omega_{M})\leq  M^{-q}\int_{\Omega_{M}} |u|^q|x|^{-\alpha}\,dx
 \leq M^{-q} \|u\|_\alpha^q\,.
 \end{equation}
 By assumption  (B2), for every
 $u\in\Phi^{-1}(-\infty,\rho]$, we have the following estimate:
 \begin{align*}
 \Psi(u)&=\int_\Omega F(x,u)\,dx\leq
 \sup_{|u|\geq M}\mathcal{F}(u)|u|^{-p}
 \int_{\Omega_{M}} |u|^p|x|^{-\alpha}
 +\int_{\Omega\setminus \Omega_{M}}F(x,u)\,dx\\
 &\leq \sup_{|u|\geq M}
 \mathcal{F}(u)|u|^{-p}\|u\|_\alpha^p\mu(\Omega_{M})^{1-p/q}
 +\sup_{|u|\leq M} \mathcal{F}(u)\mu(\Omega).
 \end{align*}
 Combining (\ref{eq3.2}) and (\ref{eq3.3}), we obtain
 $\Psi(u)\leq\Pi(F;M)$ for every $u\in \Phi^{-1}(-\infty,\rho]$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.1}]
 To apply  Bonanno's three critical points theorem, we have to verify
all conditions in  Lemma \ref{lem2.2}.

Recalling the definition of $\Phi(u),\Psi(u)$, we conclude
 that $\Phi(0)=\Psi(0)=0$ and $\Phi(u)\geq0$ for all $u\in X$,
 which is the condition {\bf (D1)} in Lemma \ref{lem2.2}.

Put $\gamma=+\infty$,  then Lemma \ref{lem2.4} and Lemma \ref{lem2.5}
 imply that the functional $I(u)=\Phi(u)-\lambda\Psi(u)$ is
 sequentially weakly lower semicontinuous and satisfies the P.-S.
 condition. Moreover, using {\rm (A4)}, (\ref{eq2.4})
 and Lemma \ref{lem2.1}, we compute
 \begin{align*}
 \Phi(u)-\lambda\Psi(u)
&\geq a_2\|u\|^p_X-
 \lambda\int_\Omega(\epsilon|u|^p|x|^{-\beta_2^*}
 +C_\epsilon|u||x|^{-\alpha})\,dx\\
&\geq a_2\|u\|^p_X-\epsilon \lambda S_{\beta^*_2}^{-1}\|u\|^p_X
 -C_\epsilon \lambda S_{1,\alpha}^{-1/p}\|u\|_X,
 \end{align*}
 fix a positive $\epsilon$ less than $a_2\lambda^{-1}S_{\beta^*_2}/2$,
 then (\ref{eq2.2}) is obvious and we manifest assumption (D3).

In the following, we verify the condition  (D2), or equivalently,
 (\ref{eq2.1}). In fact,
we can define a function the same as in \cite{Alexandru}:
\begin{equation}  \label{eq3.4}
 u_\sigma(x)= \begin{cases}
 0, &x\in \mathbb{R}^N\setminus E;\\
 K, &x\in B(x_0,\sigma r);\\
 { K\over r(1-\sigma)}(r-|x-x_0|), &x\in E\setminus
 B(x_0,\sigma r),
 \end{cases}
\end{equation}
 where $0<\sigma<1$ to be determined later. Owing to assumption
 (\ref{C1}) and (B2), we observe that
 \begin{align*}
 \Psi(u_\sigma )&=\int_E F(x,u_\sigma )dx\\
&\geq \int_{E\cap\{u_\sigma (x)=K\}}F(x,u_\sigma )dx
-\max_{|u|\leq |K|}\mathcal{F}(u)\int_{E\cap\{|u_\sigma (x)|<|K|\}}
|x|^{-\alpha}\,dx \\
&\geq \inf_{x\in E}F(x,K)\int_{B(x_0,\sigma r)}\,dx
 -\max_{|u|\leq |K|}\mathcal{F}(u)\int_{E\setminus B(x_0,\sigma r)}
 |x|^{-\alpha}\,dx.
\end{align*}
As $\sigma\to1^-$, the first term on the right hand side of the above
inequality tends to the positive constant $\omega r^N\inf_EF(x,K)$,
here $\omega$ is the volume of the unit ball,
and the second term goes to zero.
We thus pick up some $\sigma$ and $u_\sigma $ such that
$\Psi(u_\sigma )>0$. Furthermore,
 from assumption {\rm (A4)}, we see that
 $\Phi(u_\sigma )\geq a_2\|u_\sigma \|_X^p>0$.

According to the Lemma \ref{lem3.6},
to verify (\ref{eq2.1}), it suffice to turn up two positive numbers
$M$ and $\rho$, such that
 \begin{equation} \label{eq3.5}
 0<\rho<\Phi(u_\sigma )\quad\mbox{and}\quad
{\Pi(F;M)\over\rho}<{\Psi(u_\sigma )\over\Phi(u_\sigma )}.
 \end{equation}

Indeed, in view of assumption (\ref{C2}) and (B2), we see that,
 for any $\varepsilon>0$, there exist some positive constant
 $M$ such that $ \mathcal{F}(u)\leq\varepsilon |u|^p$, for all
$u\in[-M,M]$  and $\mathcal{F}(u)|u|^{-p}\leq C$ for all
$u\in \mathbb{R}$, where $C$ is independent
 of $M$. Put $\rho=\delta^p M^p$ with $\delta$ is a positive number
to be determined later,
 then we deduce
 $$
 {\Pi(F;M)\over\rho}\leq C\delta^{q-p}
 \Big({1\over a_2 S_\alpha}\Big)^{q/p}
 +\varepsilon\delta^{-p}\mu(\Omega)
 $$
One can first fix $\delta>0$ small enough, then choose $\varepsilon>0$ so small
that the right hand side of the above
inequality is less than $\Psi(u_\sigma)/\Phi(u_\sigma)$, finally
 choose $M$ and $\rho$ satisfy (\ref{eq3.5}), which yields
 condition (\ref{eq2.1}). Hence, we testify  all the conditions
in Lemma \ref{lem2.2} and
the desired conclusion follows.
\end{proof}



\begin{proof}[Proof of Theorem \ref{thm1.2}]
 Similar to the proof of Theorem \ref{thm1.1},
 denote $u_\sigma$ as (\ref{eq3.4}) and fix $\sigma =1/2$.
Owing to assumptions (\ref{C3b}) and (\ref{C4}), it is clear that
\[
 \Psi(u_\sigma)
 \geq\int_{E\cap\{u_\sigma(x)=K\}}F(x,u_\sigma)dx
\geq  c_2(K)(1+|K|^p)\int_{B(x_0,r/2)}\,dx.
\]
Moreover, recalling assumptions (A4) and  (A2), we have
 \begin{equation}  \label{eq3.11}
\begin{gathered}
 \Phi(u_\sigma) \geq a_2\int_E|\nabla u_\sigma|^p|x|^{-\beta}\,dx
 \geq a_2\Big({2|K|\over r}\Big)^p \int_{E\setminus B(x_0,r/2)}
 |x|^{-\beta}\,dx, \\
 \Phi(u_\sigma)\leq a_1\int_E(|\nabla u_\sigma|
 +|\nabla u_\sigma|^p)|x|^{-\beta}\,dx
\leq  a_1\Big({2|K|\over r}+\Big({2|K|\over r}\Big)^p\Big)
 \int_E|x|^{-\beta}\,dx.
\end{gathered}
\end{equation}
We thus get
\begin{equation} \label{eq3.6}
 \rho {\Psi(u_\sigma)\over\Phi(u_\sigma)}\geq\delta c_2(K)\rho,
\end{equation}
 where $\delta$ is a positive constant dependent only on
 $p,\beta,N,E$ and $a_1$.

On the other hand, let $M=L$ in \eqref{eq3.1},
 according to the definition in \eqref{C3}, we obtain
 \begin{equation} \label{eq3.7}
 \Pi(F;L)\leq c_3(L){L}^{p-q}
 \Big({\rho\over a_2 S_\alpha}\Big)^{q/p}+
  c_4(L) \mu(\Omega).
 \end{equation}
Denote by
 \begin{align*}
\rho_1=\Big({\delta c_2(K)L^{q-p}(a_2S_\alpha)^{q/p}
 \over 2c_3(L)}\Big)^{p\over q-p},\quad
 \rho_2={\Phi(u_\sigma)\over2}.
 \end{align*}
Let $\rho=\min\{\rho_1,\rho_2\}$. When $\rho=\rho_1$, in view of
(\ref{eq3.6}), (\ref{eq3.7}) and
 assumption (\ref{C4}), we compute
 \begin{align*}
 \rho {\Psi(u_\sigma)\over\Phi(u_\sigma)}-\Pi (F;L)
 &\geq \delta c_2(K) \rho_1-\Pi (F;L) \\
&\geq {\delta\over2} c_2(K) \rho_1 -c_4(L)\mu(\Omega)\\
&=  \delta^* (c_2(K))^{q\over q-p}L^p
 (c_3(L))^{p\over p-q}-c_4(L)\mu(\Omega) \\
&\geq  \delta^* C^{q\over q-p}c_4(L)-c_4(L)\mu(\Omega) >0,
 \end{align*}
 where $\delta^*$ and $C$ are constants dependent only on
$p,\beta,\alpha,N,E, \Omega, a_1$ and $a_2$.

In the other case of $\rho=\rho_2$, owing to (\ref{eq3.11}),
 (\ref{eq3.6}), (\ref{eq3.7}) and assumption (\ref{C4}),
 we deduce
 \begin{align*}
 \rho{\Psi(u_\sigma)\over\Phi(u_\sigma)}-\Pi(F;L)
 &\geq {\delta\over2}c_2(K)\rho_2 -c_4(L)\mu(\Omega)\\
&\geq  \delta^{**} c_2(K)|K|^p-c_4(L)\mu(\Omega)\\
&\geq \delta^{**} Cc_4(L)-c_4(L)\mu(\Omega)
 >0,
 \end{align*}
 where $\delta^{**},C$ are constants dependent only on
$p,\beta,\alpha,N,E,  \Omega, a_1$ and $a_2$. So, we achieve
assumption (\ref{eq2.1}) in any cases
 and the conclusion in the theorem is derived from Lemma \ref{lem2.2}.
\end{proof}

In the following, we give two simple examples:

\begin{example} \label{exa3.1} \rm
 Consider the  mean curvature equation
 \begin{equation} \label{eq3.8}
 \begin{gathered}
 -\mathop{\rm div}(|x|^{-\beta}(1+|\nabla u|^2)^{p-2\over2}\nabla u)
 =\lambda |u|^{m+{p-m\over |u|+1}}|x|^{-\alpha}, \quad x\in\Omega,\\
 u=0,\quad x\in\partial\Omega\,.
 \end{gathered}
 \end{equation}
Employing Theorem \ref{thm1.1}, we can get the following result:
 If $2\leq p<N$, $m<p-1,0\leq \beta<N-p$,
 $\beta_1^*\leq\alpha<\beta_2^*$, then (\ref{eq3.8}) admits
at least three distinct weak solutions.
\end{example}

\begin{example} \label{exa3.2} \rm
 Consider the $p$-Laplacian equation involving singular weight:
 \begin{equation} \label{eq3.9}
 \begin{gathered}
 -\mathop{\rm div}(|x|^{-\beta}|\nabla u|^{p-2}\nabla u)
 =\lambda |x|^{-\alpha}g(u), \quad   x\in\Omega, \\
 u=0, \quad x\in\partial\Omega,
 \end{gathered}
 \end{equation}
where
$$
g(u)= \begin{cases}
 e^u, &u\in [-t,t],\\
 e^t, &u\in  [t,\infty),\\
 e^{-t}, &u\in (-\infty,-t].
 \end{cases}
$$
Applying Theorem \ref{thm1.2}, we conclude that:
 If $1<p<N$, $0\leq \beta<N-p$ and
$\beta_1^*\leq\alpha<\beta_2^*$, then (\ref{eq3.9}) admits
 at least three distinct weak solutions provided $t$ is sufficiently large.
\end{example}

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\end{thebibliography}


\end{document}