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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 117, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2010/117\hfil Existence of nontrivial solutions]
{Existence of nontrivial solutions for singular quasilinear
 equations with sign changing nonlinearity}

\author[J. Tyagi\hfil EJDE-2010/117\hfilneg]
{Jagmohan Tyagi}

 \address{Jagmohan Tyagi\newline
 TIFR Centre For Applicable Mathematics\\
 Post Bag No.-6503, Sharda Nagar, \newline
 Chikkabommasandra, Bangalore-560065, Karnataka, India}
 \email{jagmohan.iitk@gmail.com, jtyagi1@gmail.com}

\thanks{Submitted April 15, 2010. Published August 20, 2010.}
\subjclass[2000]{35J92, 35J75}
\keywords{p-Laplacian; singularity; multiple solutions}

\begin{abstract}
 By an application of Bonanno's three critical point theorem,
 we establish the  existence of a nontrivial solution to the problem
 \begin{gather*}
 -\Delta_p u= \mu \frac{g(x)|u|^{p-2}u}{|x|^p}
 +\lambda a(x)f(u) \quad \text{in }\Omega,\\
  u =0\quad \text{on }\partial \Omega,
 \end{gather*}
 under some restrictions on $g,a$ and $f$ for certain positive
 values of $\mu$ and $\lambda$.
\end{abstract}

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

\section{Introduction}

 Let us set up a problem which is used to give a brief introduction
about previous research
\begin{equation} \label{Pgmla} %(P_{g,\mu,\lambda,a})
\begin{gathered}
-\Delta_p u= \mu \frac{g(x)|u|^{p-2}u}{|x|^p}
+ a(x)f_1(\lambda,u)\quad \text{in }\Omega, \\
u =0\quad \text{on }\partial \Omega,
\end{gathered}
\end{equation}
where $\Omega$ be a bounded domain in $\mathbb{R}^N$
with smooth boundary $\partial \Omega$, $0 \in \Omega$ and
$f_1: (0,\infty)\times\mathbb{R} \to \mathbb{R}$
is a continuous function. Suppose there exists $ M>0$ such
that  $-M \leq g(x) \leq 1,a \in L^{\infty}(\Omega)$ and
$0\leq \mu < (\frac{N-p}{p})^p$. Let $\lambda$ be a positive parameter.

In the last few years, problem \eqref{Pgmla}
with  $\mu=0$ and $a(x)\equiv 1$ has been extensively investigated
for the case $p=2$, (see,\,\cite{guo,hai,saint}
and the references cited therein), where $f_1(\lambda,u)= \lambda f(u)$.
In case $p=2$, there are many publications dealing with the existence
of solution to the problem \eqref{Pgmla} with $g=1$ and $a=1$.
For convenience of the reader, we give a brief summary of these
results. Ferrero and Gazzola \cite{ferrero}  considered the
problem \eqref{Pgmla}, where $f_1(\lambda,u)= |u|^{2^*-2} u+ \lambda u.$
They established the existence of nontrivial solution by variational
method for certain values of $\mu$ and $\lambda$.
Ruiz and Willem \cite{ruiz}  considered the aforesaid problem,
where $f_1(\lambda,u)= |u|^{2^*-1} u+ \lambda u$  and established the
existence of  positive solutions under various assumptions on
the domain $\Omega$. Chen \cite{chen1,chen2} also studied the
same problem and obtained multiple solutions by analyzing the
exact growth order of the positive solutions near origin,
where $f_1(\lambda,u)= u^{2^*-1}_{+} + \lambda u^q_+,0<q<1,$
$\lambda >0$, $0\leq \mu<(N-2)^2/4$.
Recently, Krist\'{a}ly  and Varga \cite{kristaly} obtained
the existence of three solutions to the problem
\eqref{Pgmla} with $g=1$ and $a=1,$ by an application of Bonanno's three
critical point theorem \cite{bona},  where $f_1(\lambda,u)=\lambda f(u).$

For a good amount of work concerning quasilinear equations with
singularities, we refer the book of Drabek et al.\,\cite{drabek}
and for existence and multiplicity results concerning singular
p-Laplacian, we refer the reader to\,\cite{faraci,monte,yang}
and reference cited therein.  Montefusco\,\cite{monte} considered
the problem \eqref{Pgmla} with $g=1$ and $a=1$,
where $f_1(\lambda,u)= |u|^{q-2} u,1<p<q<p^*,1<p<N$. He established
the existence of a nontrivial solution whenever
$\mu \in (0,(\frac{(N-p)}{p})^p)$ is fixed. Faraci
and Livrea \cite{faraci} utilized Montefusco's result and
gave some bifurcation results for singular p-Laplacian.
By an application of Bonanno's three critical point theorem,
Yang et al. \cite{yang} established the existence of three
weak solutions to singular p--Laplacian type equation,
which has singularity in the principal part of the operator.

In this study, our main purpose inspired by \cite{kristaly},
is to see that the conditions introduced by \cite{kristaly} on $f$
can be extended for singular p-Laplacian with sign-changing
nonlinearity also.
It is worth noting that to establish the existence of solutions
to the problem \eqref{Pgmla} is of more interest due
to the presence of singular potential as well as sign changing
nonlinearity.
In this note, we establish the existence of two solutions to
the problem \eqref{Pgmla} by  Bonanno's theorem, where
$f_1(\lambda,u)= \lambda f(u)$. More precisely, we give the existence
of two solutions to the problem
\begin{equation}
\begin{gathered}
-\Delta_p u= \mu \frac{g(x)|u|^{p-2}u}{|x|^p}+\lambda a(x)f(u) \quad
\text{in }\Omega,\\
  u =0\quad \text{on }\partial \Omega, \label{e1}
\end{gathered}
\end{equation}
where $f\in C(\mathbb{R},\mathbb{R})$ and satisfies the following hypotheses:
\begin{itemize}
\item[(H1)] $\lim_{s\to 0}\frac{f(s)}{s^{p-1}}=0$.

\item[(H2)] $\lim_{|s|\to \infty}\frac{f(s)}{|s|^{p-1}}=0$.

\item[(H3)] Let $F(s)= \int_{0}^{s}f(t) dt$, we assume
 $ \sup_{s\in \mathbb{R}} F(s) >0$.
\end{itemize}
We state now the theorem we will prove in Section 4.

\begin{theorem} \label{thm1.1}
 Let $f \in C(\mathbb{R},\mathbb{R})$ which satisfies the 
 hypotheses {\rm (H1)--(H3)}.
Let there exists $M>0$ such that
$-M\leq g(x)\leq 1,a \in L^{\infty}(\Omega)$.
Then for every $\mu \in[0,(\frac{N-p}{p})^p)$ there exist an open
interval $\Lambda_{\mu} \subset (0,\infty)$ and a real number
$\eta_\mu>0$ such that for every $\lambda \in \Lambda_{\mu}$,
the problem \eqref{e1} has one non-trivial weak solution
$u \in W_{0}^{1,p}(\Omega)$ such that
$\|u \|_{W_{0}^{1,p}(\Omega)} \leq \eta_\mu$.
\end{theorem}

We remark that in proving the above theorem
ideas from \cite{kristaly} are used.
We organize this paper as follows:
Section 2 deals with the preliminaries.
Section 3 deals with some lemmas which have been used
in the main theorem. The main result is proved in Section 4.
In the last section, we construct some examples for the
illustration of main result.

\section{preliminaries}

Let $\Omega$ be a bounded domain in $\mathbb{R}^N$  with a smooth
boundary $\partial \Omega$  and $0 \in \Omega$. The space
$W_{0}^{1,p}(\Omega)$ is endowed by the norm
$$
\|u\|_{W_{0}^{1,p}}= \Big(\int_{\Omega} |\nabla u|^p \Big)^{1/p}.
$$
Let $1<p<N$, we recall classical Hardy's inequality, which says that
\begin{equation} \label{HS}
\int_{\Omega} \frac{|u(x)|^p}{|x|^p} dx
\leq \frac{1}{C_{N,p}}\int_{\Omega}|\nabla u|^p,\quad
u \in W_{0}^{1,p}(\Omega),
\end{equation}
where $C_{N,p}= (\frac{N-p}{p})^p$. For a detail about Hardy inequality
and related problem, we refer the reader to \cite{garcia}.
The Hardy inequality proves that embedding of $ W_{0}^{1,p}(\Omega)$
in $L^{p}(\Omega,\frac{1}{|x|^p})$ is continuous but is not compact
as for the Sobolev embeddding. The Sobolev embedding constant of
the compact embedding
$W_{0}^{1,p}(\Omega)\circlearrowleft L^{q}(\Omega),q \in [1,p^*)$,
will be denoted by $c(N,p)>0$;
i.e.,$\|u\|_{W_{0}^{1,p}}\geq c(N,p)\|u\|_{L^{p}},$\,for every
$u \in W_{0}^{1,p}$.
Let us define $F(s)= \int_{0}^{s} f(t) dt$. We introduce the energy
functional $E_{\mu,\lambda}: W_{0}^{1,p}(\Omega) \to \mathbb{R}$
associated with \eqref{e1},
$$
E_{\mu,\lambda}= \Phi_{\mu}(u)-\lambda J(u),u \in  W_{0}^{1,p}(\Omega),
$$
where
$$
\Phi_{\mu}(u)=\frac{1}{p}\int_{\Omega}|\nabla u|^p dx
- \frac{\mu}{p}\int_{\Omega} \frac{g(x)|u(x)|^p}{|x|^p} dx,J(u)
= \int_{\Omega} a(x)F(u(x)) dx.
$$
It is easy to see that the critical points of $E_{\mu,\lambda}$
are exactly the weak solutions of \eqref{e1}. Therefore, it
is sufficient to give the existence of multiple critical points
of $E_{\mu,\lambda}: W_{0}^{1,p}(\Omega)\to \mathbb{R}$ for certain values
of $\mu$ and $\lambda$. To establish the existence of critical
points of $E_{\mu,\lambda}$, we use Bonanno's three critical
point theorem. Since Bonanno's result \cite{bona} is a
special case of Ricceri's three critical point theorem
\cite{ricceri}, so for the reader's convenience we give a
brief sketch.

Ricceri \cite{ricceri} proved the following result.

\begin{theorem}\label{thm1}
Let $X$ be a separable and reflexive real Banach space,
$I\subset \mathbb{R}$ an interval, and $g: X \times I\to \mathbb{R}$
 a continuous function satisfying the following conditions:
\begin{itemize}
\item[(i)] for each $x\in X,g(x,.)$ is continuous and concave;

\item[(ii)] for each $\lambda \in I,g(.,\lambda)$ is sequentially weakly
 lower semicontinuous and  G\^{a}teaux differentiable and
$$
\lim_{\|x\|\to \infty} g(x,\lambda) =+\infty;
$$

\item[(iii)] there exists a continuous concave function
 $h: I\to \mathbb{R}$ such that
$$\sup_{\lambda \in I}\inf_{x\in X} (g(x,\lambda)+ h(\lambda))
< \inf_{x \in X} \sup_{\lambda \in I} (g(x,\lambda)+ h(\lambda)).
$$
\end{itemize}
Then there exist an open interval $\Lambda \subset I$ and a
positive real number $\eta$, such that for each $\lambda \in \Lambda$,
the equation
$$
g'_{x}(x,\lambda)=0
$$
admits at least two solutions in $X$ whose norms are less than $\eta$.

If, in addition, the function $g$ is continuous in $X \times I$,
and for each $\lambda \in I$, the function $g(.,\lambda)$ is $C^1$ and
satisfies the Palais-Smale condition, then the above conclusion
holds with ``three solutions'' instead of ``two solutions''.
\end{theorem}

As a special case of the above theorem, Bonanno \cite{bona} gave the following

\begin{theorem} \label{thm2}
Let $X$ be a separable and reflexive real Banach space and
$\Phi,J: X\to \mathbb{R}$ be two continuously G\^{a}teau
differentiable functionals. Assume that there exists $x_0 \in X$
such that $\Phi(x_0)=0=J(x_0)$ and $\Phi(x) \geq 0$ for every
$x \in X$ and suppose there exist $x_1 \in X$ and $r>0$ such that
\begin{itemize}
\item[(i)] $r < \Phi(x_1)$;
\item[(ii)] $\sup_{\Phi(x)<r} J(x) < r \frac{J(x_1)}{\Phi(x_1)}$.
\end{itemize}
Further, put
$$
\bar{a}= \frac{h r }{r \frac{J(x_1)}{\Phi(x_1)}
-  \sup_{\Phi(x)< r} J(x)},
$$
with $h>1$, and assume that the functional $\Phi- \lambda J$
is sequentially weakly lower semicontinuous, satisfies
Palais-Smale condition and
\begin{itemize}
\item[(iii)] $\lim_{\|x\|\to + \infty} (\Phi(x)- \lambda J(x))= + \infty$
 for every $\lambda \in [0,\bar{a}]$.
\end{itemize}
Then there exist an open interval $\Lambda \subseteq [0,\bar{a}]$
and a positive real number $\eta$ such that for each $\lambda \in \Lambda$,
the equation
\[
\Phi'(x)-\lambda J'(x)=0
\]
admits at least three solutions in $X$ whose norms are less than $\eta$.
\end{theorem}

We remark that in view of Ricceri's theorem \cite{ricceri},
if we drop the Palais-Smale condition  and
continuous G\^{a}teau differentiability of the functional
$g(.,\lambda)= \Phi(.)-\lambda J(.)$ from Theorems \ref{thm1} and \ref{thm2},
we have the existence of two solutions. This fact is carried
out in Theorem \ref{thm1}.

\section{Auxiliary lemmas}

In this section, we sate some lemmas to be used in
the proof of main theorem.

\begin{lemma}\label{lm1}
 For every $\mu \in [0,C_{N,p})$ and $\lambda \in \mathbb{R}$,
the functional $E_{\mu,\lambda}$ is coercive.
\end{lemma}

\begin{proof}
Let us fix $\mu \in [0,C_{N,p})$ and $\lambda \in \mathbb{R}$ be arbitrary.
By (H2), for any given $\lambda \in \mathbb{R}$, there exists
$\delta=\delta(\mu,\lambda)>0$ such that
$$
|f(s)| <  \big(1-\frac{\mu}{C_{N,p}}\big)
\frac{c(N,p)^{-p}} {(1+ \|a\|_{L^{\infty}})} (1+|\lambda|)^{-1} |s|^{p-1},
$$
whenever $|s|> \delta$. This implies
$$
|f(s)| <  \big(1-\frac{\mu}{C_{N,p}}\big)
\frac{c(N,p)^{-p}} {(1+ \|a\|_{L^{\infty}})} (1+|\lambda|)^{-1} |s|^{p-1}
+ \max_{|t|\leq \delta}|f(t)|,\forall\,s \in \mathbb{R}.
$$
An integration yields,
\begin{equation} \label{2}
|F(s)| < \frac{1}{p} \big(1-\frac{\mu}{C_{N,p}}\big)
 \frac{c(N,p)^{-p}} {(1+ \|a\|_{L^{\infty}})} (1+|\lambda|)^{-1} |s|^{p}
+ \max_{|t|\leq \delta} |f(t)| |s|,\forall s \in \mathbb{R}.
\end{equation}
Since we have
$$
E_{\mu,\lambda}(u)= \frac{1}{p}\int_{\Omega}|\nabla u|^p dx
- \frac{\mu}{p}\int_{\Omega} \frac{g(x)|u(x)|^p}{|x|^p} dx
-\lambda \int_{\Omega} a(x)F(u(x)) dx,
$$
so by  Hardy inequality, for every $u \in W_{0}^{1,p} $ and
using the fact that $-M \leq g(x)\leq 1$, we have
\begin{equation} \label{3}
\begin{aligned}
 E_{\mu,\lambda}(u)
&\geq \frac{1}{p}\int_{\Omega}|\nabla u|^p dx
  - \frac{\mu}{p}\,\int_{\Omega} \frac{|u(x)|^p}{|x|^p}| dx
  -\lambda \int_{\Omega} a(x)F(u(x)) dx\\
&\geq \frac{1}{p}\int_{\Omega}|\nabla u|^p dx
  - \frac{\mu}{ C_{N,p}\,p}\int_{\Omega} |\nabla u|^p  dx
  -\lambda \int_{\Omega} a(x)F(u(x)) dx\\
& \geq \frac{1}{p}\big(1-\frac{\mu}{C_{N,p}} \big)
  \int_{\Omega}|\nabla u|^p dx
  - |\lambda| \int_{\Omega}|a(x)\|F(u(x))| dx\\
& \geq \frac{1}{p}\big(1-\frac{\mu}{C_{N,p}} \big)
  \int_{\Omega}|\nabla u|^p dx\\
&\quad - \frac{|\lambda|}{(1+ |\lambda|)p}c(N,p)^{-p}
   \int_{\Omega} |u|^{p} dx - |\lambda|c(N,1)
  \max_{|t|\leq \delta}|f(t)\||u\|_{ W_{0}^{1,p}}
  \\
& \geq \frac{1}{p}\big(1-\frac{\mu}{C_{N,p}} \big)
  \big(\frac{1}{1+ |\lambda|}\big) \int_{\Omega}|\nabla u|^p dx
   - |\lambda|c(N,1) \max_{|t|\leq \delta}|f(t)\||u\|_{ W_{0}^{1,p}}\\
& \geq \frac{1}{p}\big(1-\frac{\mu}{C_{N,p}} \big)
  \big(\frac{1}{1+ |\lambda|}\big)\|u\|^{p}_{ W_{0}^{1,p}}
  - |\lambda|c(N,1) \max_{|t|\leq \delta}|f(t)\||u\|_{ W_{0}^{1,p}},
\end{aligned}
\end{equation}
where we have used  \eqref{2}.
Now if $\|u\|_{W_{0}^{1,p}}\to \infty$, one can conclude that
$E_{\mu,\lambda} \to \infty$ and hence $E_{\mu,\lambda}$
is coercive.
\end{proof}

\begin{lemma}\label{lem2}
Assume that $\mu \in [0,C_{N,p}]$, then $\Phi_{\mu}(u)$ is a
sequentially weakly lower semicontinuous functional on
$W_{0}^{1,p}(\Omega)$.
\end{lemma}

\begin{proof}
Montefusco \cite{monte}, proved the sequentially weakly
lower semicontinuity of the functional
$$
\Phi(u)= \frac{1}{p}\int_{\Omega}|\nabla u|^p dx
- \frac{\mu}{p}\int_{\Omega} \frac{|u(x)|^p}{|x|^p} dx,
$$
using the ideas from Lions \cite{lions1,lions2}.
Since  for $-M\leq g(x) \leq 1$, the proof of this lemma
is similar to the proof of \cite[Theorem 3.2]{monte},
so we omit the details.
\end{proof}

\begin{lemma}\label{lem3}
For every $\mu \in [0,C_{N,p})$ and $\lambda \in \mathbb{R}$, the functional
$E_{\mu,\lambda}$ is sequentially weakly lower semicontinuous
functional on $W_{0}^{1,p}(\Omega)$.
\end{lemma}

\begin{proof}
By Lemma \ref{lem2}, $\Phi_{\mu}(u)$ is a sequentially weakly
lower semicontinuous functional on $W_{0}^{1,p}(\Omega)$, for all
$\mu \in [0,C_{N,p})$.
By  (H2), there exists $C>0$ such that
\begin{equation} \label{4}
|f(s)| \leq C(1+|s|^{p-1}),\quad s\in \mathbb{R}.
\end{equation}
Now the sequentially weak continuity of $J$ is obtained by
a classical way. So, this proves the lemma.
\end{proof}

\begin{lemma}\label{lem4}
For every $\mu \in [0,C_{N,p})$,
$$
\lim_{\xi\to 0^+}  \frac{\sup \{J(u): \Phi_{\mu}(u)<\xi \}} {\xi} =0.
$$
\end{lemma}

\begin{proof}
We fix $\mu \in [0,C_{N,p})$. By  (H1), for any given
 $\epsilon>0$ there exists a $\delta(\epsilon)$ such that
\begin{equation} \label{5}
|f(s)| < \frac{\epsilon}{2}\big( 1-\frac{\mu} {C_{N,p}} \big)
\frac{c(N,p)^{-p}} {(1+ \|a\|_{L^{\infty}})}|s|^{p-1},\quad
\text{whenever }|s|<\delta.
\end{equation}
We fix a $\gamma_1 \in (p,p^*)$ and combining  \eqref{4} and \eqref{5}
yields
\begin{equation} \label{6}
|F(s)|\leq \frac{\epsilon}{2p}\big( 1-\frac{\mu}{C_{N,p}} \big)
\frac{c(N,p)^{-p}} {(1+ \|a\|_{L^{\infty}})} |s|^{p}
+ C \frac{(1+ \delta)}{(1+ \|a\|_{L^{\infty}})}
\delta^{1-\gamma_1}|s|^{\gamma_1},
\end{equation}
for all $s\in \mathbb{R}$.
For $\xi>0$, we define the sets
$$
A_{\xi} = \{u\in W_{0}^{1,p}:\Phi_{\mu}(u)< \xi \};\quad
B_{\xi} = \{u\in W_{0}^{1,p}:\big( 1-\frac{\mu}{C_{N,p}}\big)
 \|u\|^{p}_{W_{0}^{1,p}} < \xi\,p\}.
$$
By an application of \eqref{HS}, one can observe that
$A_{\xi} \subseteq B_{\xi}$. By \eqref{6},for every
$u\in A_{\xi}$ and hence  $u\in B_{\xi}$ we have
\begin{equation} 
\begin{aligned}
& J(u)\\
&\leq \frac{\epsilon}{2p}\big( 1-\frac{\mu}{C_{N,p}}\big)
  c(N,p)^{-p} \int_{\Omega} |u|^{p} dx
 + C (1+ \delta) \delta^{1-\gamma_1} \int_{\Omega} |u(x)|^{\gamma_1} dx\\
& \leq \frac{\epsilon}{2p}\big(1-\frac{\mu}{C_{N,p}} \big)
  \int_{\Omega}|\nabla u|^p dx +C(1+\delta)^{1-\gamma_1}
  c(N,\gamma_1)^{\gamma_1} p^{\gamma_1/p} \xi^{\gamma_1/p}
 \Big(1-\frac{\mu}{C_{N,p}}\Big)^{-\gamma_1/p} \\
& \leq \frac{\epsilon}{2} \xi+ C(1+\delta)^{1-\gamma_1}
 c(N,\gamma_1)^{\gamma_1} p^{\gamma_1/p} \xi^{\gamma_1/p}
  \Big(1-\frac{\mu}{C_{N,p}}\Big)^{-\gamma_1/p} \\
& \leq \frac{\epsilon}{2} \xi+C_1  \xi^{\gamma_1/p},
\end{aligned}
\end{equation}
where
\[
C_1= C(1+\delta)^{1-\gamma_1} c(N,\gamma_1)^{\gamma_1} p^{\gamma_1/p}
\Big(1-\frac{\mu}{C_{N,p}}\Big)^{-\gamma_1/p}.
\]
Thus there exists $\xi(\epsilon)>0$ such that for every
$0<\xi<\xi(\epsilon)$,
$$
0\leq \frac{\sup_{u \in A_{\xi} }  J(u)}   {\xi}
\leq  \frac{\sup_{u \in B_{\xi} }  J(u)}  {\xi}
\leq \frac{\epsilon}{2} + C_1 \xi^{\frac{\gamma_1 -p}{p}}<   \epsilon ,
$$
which proves the lemma.
\end{proof}

Now we are ready to sketch the proof of the main result.

\section{Proof of Theorem \ref{thm1.1}}

\begin{proof}
Let $t_0 \in \mathbb{R}$ such that $F(t_0)>0$, by  (H3). We choose
 $R_0>0$ such that $R_0< \operatorname{dist}(0,\partial \Omega)$.
For $\eta \in (0,1)$ as already defined in \cite{kristaly},
we also define
\[
u_{\eta}(x)= \begin{cases}
  0,    &\text{if } x \in \mathbb{R}^N  \setminus B_N(0,R_0); \\
  t_0,  &\text{if } x \in B_N(0,\eta R_0); \\
  \frac{t_0}{R_0 (1-\eta)}(R_0-|x|),  &\text{if }
 x \in B_N(0,R_0) \setminus B_N(0,\eta R_0),
\end{cases}
\]
where $B_N(0,r)$ denotes the $N$-dimensional open ball with
center $0$ and radius $r>0$. It is easy to see that
$u_{\eta} \in W_{0}^{1,p}$. Let $V_{N}$ denote the volume of
the $N$-dimensional unit ball in $\mathbb{R}^{N}$, one can compute
\begin{equation} \label{t1}
 \|u_{\eta}\|^{p}_{W_{0}^{1,p}}
= t_0^p R_0^{N-p}(1-\eta)^{-p} V_N (1-\eta^{N})
\end{equation}
and
\begin{equation} \label{t2}
J(u_{\eta})\geq [F(t_0) \eta^N- \max_{|t|\leq |t_0|}
|F(t)|(1-\eta^{N})]V_N R_0^{N}.
\end{equation}
For $\eta$ close enough to $1$, the right hand side of the
last inequality becomes strictly positive, so we choose such
a number, say $\eta_0$. We fix $\mu \in [0,C_{N,p})$.
By Lemma \ref{lem4} and  in view of \eqref{t1}, we may
choose $\xi_0$ such that
\begin{gather*}
  p \xi_{0}< \big(1- \frac{\mu} {C_{N,p}}\big)
\|u_{\eta_0}\|^{p}_{W_{0}^{1,p}}, \\
\sup\{J(u):\Phi_{\mu}(u)< \xi_0\}
<\frac{ p [F(t_0) \eta^N- \max_{|t|\leq |t_0|} |F(t)|(1-\eta^N)]
V_{N} R_0^N} {\|u_{\eta_0}\|^{p}_{W_{0}^{1,p}}}.
\end{gather*}
By choosing $x_1=u_{\eta_0}$, hypotheses of Theorem \ref{thm2}
are satisfied. Define
\begin{equation} \label{a}
 \bar{A}=\bar{A_\mu}=\frac{1+\xi_0} { \frac{J(u_{\eta_0})}
 {\Phi_{\mu}(u_{\eta_0})}- \frac{\sup\{J(u):
 \Phi_{\mu}(u)< \xi_0\}} {\xi_0}}.
\end{equation}

In view of Lemmas \ref{lm1}, \ref{lem3}, all the hypotheses
of Theorem \ref{thm2} are satisfied after putting $x_0=0$.
An application of Theorem \ref{thm2} implies that there exist
an open interval $\Lambda_\mu \subset [0,\bar{A_\mu}]$ and a
number $\eta_\mu>0$ such that for each $\lambda \in \Lambda_\mu$,
 the equation $E'_{\mu,\lambda}\equiv \Phi'_{\mu}(u)-\lambda J'(u)=0$,
admits at least two solutions in $W_{0}^{1,p}$ which have
$W_{0}^{1,p}$-norm less than $\eta_\mu$. Since  (H1)
implies that $f(0)=0$, so  {\rm (H1)} admits one trivial
solution and hence there exists a nontrivial solution to \eqref{e1},
 which completes the proof.
\end{proof}

\begin{remark} \label{rmk4.1} \rm
Let $g(x)\equiv 1 \equiv a(x)$ and $p=2$ in \eqref{e1},
then the proof of this corollary is given by Krist\'{a}ly
and Varga \cite{kristaly}. In fact, they obtained the existence
of three solutions. Since in the present study, $E_{\mu,\lambda}$
fails to satisfy the Palais-Smale condition, so we get
the existence of two solutions.
\end{remark}

\begin{remark} \label{rmk4.2} \rm
As in \cite{kristaly}, we also give the explicit estimation of the
interval $\Lambda_\mu, \,\mu \in [0,C_{N,p})$.
We fix $t_0,R_0,\eta_0$ as in the previous section. In view of
Lemma \ref{lem4}, we have
$$  \frac{\sup\{J(u):\Phi_{\mu}(u)< \xi_0\}}{\xi_0}
<\frac{J(u_{\eta_0})}{2 \Phi_{\mu}(u_{\eta_0})}.
$$
Then by \eqref{a}, one can see that
$$
\Lambda_\mu \subset \big[0,\frac{4}{p}\big(1-\frac{\mu}{C_{N,p}}\big)
 \big(\frac{t_0}{R_0}\big)^p  \frac{(1-\eta_0)^{-p}(1-\eta_{0}^{N})}
{[F(t_0) \eta_{0}^N- \max_{|t|\leq |t_0|} |F(t)|(1-\eta_{0}^{N})]}
\big].                                 $$
\end{remark}

\section{Examples}

In this section, we construct some examples for the illustrations
of main theorem.
\begin{example} \label{exa5.1} \rm
Consider \eqref{e1} with $g(x)= 1-e^{-|x|^2},a(x)= \sin |x|$
and $\mu \in[0,(\frac{N-p}{p})^p)$. Suppose there exist $c>p-1$
and $S>0$ such that
$$
f(s)= \begin{cases}
  0, & s\leq 0; \\
  s^c, & 0<s\leq S; \\
  e^{-s}-e^{-S}+ S^c, & S<s.
\end{cases}
$$
Then it is easy to see that $g,a$ and $f$ satisfy the hypotheses
of Theorem \ref{thm1.1}. An application of Theorem \ref{thm1.1} gives
the existence of an open interval $\Lambda_{\mu} \subset (0,\infty)$
and a real number $\eta_\mu>0$ such that for every
$\lambda \in \Lambda_{\mu}$, the problem \eqref{e1} has one
non-trivial weak solution $u \in W_{0}^{1,p}(\Omega)$
such that $\|u \|_{W_{0}^{1,p}(\Omega)} \leq \eta_\mu$.
\end{example}

\begin{example} \label{exa5.2} \rm
Consider \eqref{e1} with
$g(x)= \frac{1+ \sin |x|}{2},a(x)= (1+|x|)^{-\alpha}\cos |x|$,
where $\alpha>0$ and $\mu \in[0,(\frac{N-p}{p})^p)$.
Suppose there exist $\beta>0,c>0$ such that $\beta <p-1<c$ and
 $S>0$ such that
$$
f(s)= \begin{cases}
  0, & s\leq 0; \\
  e^{(s^c)}-1, & 0<s\leq S; \\
  e^{(S^c)}-S^\beta+ s^\beta-1, & S<s.
\end{cases}
$$
Then it is not difficult to see that $g,a$ and $f$ satisfy
the hypotheses of Theorem \ref{thm1.1}. By
Theorem \ref{thm1.1}, \eqref{e1} has one non-trivial weak solution
$u \in W_{0}^{1,p}(\Omega)$ such that
$\|u \|_{W_{0}^{1,p}(\Omega)} \leq \eta_\mu $ for every
$\lambda \in \Lambda_{\mu}$, where the existence of
$\Lambda_{\mu} \subset (0,\infty)$ and a real number
$\eta_\mu>0$ are guaranteed by Theorem \ref{thm1.1}.
\end{example}

\begin{example} \label{exa5.3} \rm
Consider \eqref{e1} with $g(x)=1$, $a(x)= e^{\sin |x|}$ and
$\mu \in[0,(\frac{N-p}{p})^p)$. Suppose there exist $\beta>0,c>0$
such that $\beta <p-1<c$ and $S>0$ such that
$$
f(s)= \begin{cases}
  |s|^\beta- |S|^\beta, & s< -S; \\
  0, &   -S\leq s\leq 0; \\
  s^c(\sin s + e^{-s}), & 0<s\leq S; \\
  S^c (\sin S + e^{-S}), & S<s.
\end{cases}
$$
Then it is easy to see that $g,a$ and $f$ satisfy the hypotheses
of Theorem \ref{thm1.1} and hence \eqref{e1} has one non-trivial
weak solution $u \in W_{0}^{1,p}(\Omega)$ such that
$\|u \|_{W_{0}^{1,p}(\Omega)} \leq \eta_\mu $ for every
$\lambda \in \Lambda_{\mu}$, where the existence of
$\Lambda_{\mu} \subset (0,\infty)$ and a real number $\eta_\mu>0$
are guaranteed by Theorem \ref{thm1.1}.
\end{example}


\subsection*{Acknowledgments}
The author wants to thank Professors Mythily Ramaswamy and
S. Prashanth for their useful
conversations; also wants to thank the anonymous referee
for his/her constructive remarks which improved the original
manuscript considerably. Financial support
under  grant 2/40(6)/2009--R\&D--II/166 from NBHM, DAE,
Govt. of India is gratefully acknowledged.


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