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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 43, 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 2011 Texas State University - San Marcos.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2011/43\hfil Existence and multiplicity of solutions]
{Existence and multiplicity of solutions for divergence type
elliptic equations}

\author[L. Zhao, P. Zhao, X. Xie \hfil EJDE-2011/43\hfilneg]
{Lin Zhao, Peihao Zhao, Xiaoxia Xie}

\address{Lin Zhao \newline
School of Mathematics and Statistics, Lanzhou University\\
Lanzhou, Gansu 730000, China}
\email{zhjz9332003@gmail.com}

\address{Peihao Zhao \newline
School of Mathematics and Statistics, Lanzhou University\\
Lanzhou, Gansu 730000, China}
\email{zhaoph@lzu.edu.cn}

\address{Xiaoxia Xie \newline
School of Mathematics and Statistics, Lanzhou University\\
Lanzhou, Gansu 730000, China}
\email{xiexx06@lzu.cn}


\thanks{Submitted January 1, 2011. Published March 31, 2011.}
\subjclass[2000]{35A15, 35J20, 35J62}
\keywords{Nonlinear elliptic equations; uniformly convex; mountain
pass lemma; \hfill\break\indent three critical points theorem}

\begin{abstract}
 We establish the existence and multiplicity of weak solutions
 of a problem involving a uniformly convex elliptic operator
 in divergence form.
 We find one nontrivial solution by the mountain pass lemma,
 when the nonlinearity has a $(p-1)$-superlinear growth at infinity,
 and two nontrivial solutions by minimization and mountain pass
 when the nonlinear term has a $(p-1)$-sublinear growth at infinity.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}

 In this article we study the boundary-value problem
\begin{gather}
-\operatorname{div}(a(x,\nabla u))+|u|^{p-2}u=\lambda f(x,u), \quad
 x\in\Omega, \label{e1.1} \\
 u(x)=\text{constant}, \quad x\in\partial\Omega, \label{e1.2} \\
 \int_{\partial\Omega}a(x,\nabla u)\cdot n\,ds=0, \label{e1.3}
\end{gather}
where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$,
with smooth boundary.
We obtain the existence and multiplicity for the equation
\begin{equation}
-\operatorname{div}(a(x,\nabla u))
=f(x,u). \label{eP}
\end{equation}
Such operators arise, for example, from the expression of the
$p$-Laplacian in curvilinear coordinates. We refer to the
books \cite{k2,s1,z1} for the foundation of the variational methods
and refer to the overview papers \cite{b1,d1,k1,l1,l2,n1,p1,r1,y1}
for the advances and references of this area. Recently, the Dirichlet
problem \eqref{eP} was studied and obtained one weak solution
by the mountain pass lemma in \cite{n1}, when the potential
satisfies a set of assumptions and $f$ is $(p-1)$-superlinear
at infinity. Duc and Vu \cite{d1} extended the result of \cite{n1},
considering the Dirichlet problem \eqref{eP} in the nonuniform case.
Krist\'aly, Lisei and Varga \cite{k1}
study the Dirichlet problem \eqref{eP},
and obtain three solutions when $f$ is $(p-1)$-sublinear at infinity.
Yang, Geng and Yan \cite{y1} deal with the singular $p$-Laplacian type
equation and get three solutions with $f$ having $(p-1)$-sublinear
growth at infinity. Papageorgiou, Rocha and Staicu \cite{p1} consider
the nonsmooth $p$-Laplacian problem, and obtain at least two solutions.
In \cite{l2}, the sub-supersolution method has been applied to find
one solution to the problem \eqref{eP} with the boundary
condition \eqref{e1.2} and \eqref{e1.3} where the nonlinearity $f$
satisfies the condition: $|f(x,u)|\leq a_3(x)$, with
$a_3 \in L^{p'}(\Omega)$, $\frac{1}{p}+\frac{1}{p'}=1$.

The first result of this paper is about the existence of solution
of \eqref{e1.1}-\eqref{e1.3}. We assume that the nonlinear term
$f:\Omega\times \mathbb{R}\to \mathbb{R}$ satisfies
the Ambrosetti-Rabinowitz type condition and obtain one weak
solution by the mountain pass lemma in Theorem \ref{thm3.1}.

The second result of this paper is about the existence and
multiplicity of solutions for the problem \eqref{e1.1}-\eqref{e1.3}.
Under the growth on $f$, saying, $f$ is $(p-1)$-sublinear at infinity,
we obtain two nontrivial solutions by minimization and mountain pass
lemma in \cite{b1,k1,p1}, where they do the same thing under different
assumptions on $f$.

We remark that in \cite{d1,k1,n1}, the function $A$, with
$\nabla_{\xi}A=a(x,\xi)$, satisfies the $p$-uniformly convex
condition: there exists a constant $k>0$ such that
\[
 A(x,\frac{\xi+\eta}{2})\leq\frac{1}{2}A(x,\xi)
+\frac{1}{2}A(x,\eta)-k|\xi-\eta|^{p}, \quad x\in \Omega,\;
\xi,\eta\in\mathbb{R}^{N}.
\]
However, for the case $A(\xi)=|\xi|^{p}$, the $p$-uniform convexity
condition is satisfied only for $p\in [2,+\infty)$.
We assume the function $A$ satisfies the condition (UC) in this paper,
while the condition (UC) is satisfied for $A(\xi)=|\xi|^{p}$
for all $p\in(1,+\infty)$ (see \cite{f1}).


\section{Preliminaries}

 Let $X$ be a Banach space and $X^{\ast}$ is its topological
dual.  We denote the duality brackets for
the pair $(X^{\ast},X)$ by $\langle\cdot,\cdot\rangle$ and
$W^{1,p}(\Omega)$ $(p>1)$ is the usual Sobolev space,
equipped with the norm
\begin{equation}
\|u\|=\|u\|_{W^{1,p}(\Omega)}=\Big(\int_{\Omega}|\nabla
u|^{p}+|u|^{p}dx\Big)^{1/p}. \label{e2.1}
\end{equation}
Let
\[
V=\{u\in W^{1,p}(\Omega):u|_{\partial\Omega}=\text{constant}\}.
\]
We next claim that $V$ is a closed subspace of $W^{1,p}(\Omega)$
and thus a reflexive Banach space with the
restricted norm of \eqref{e2.1}.

\begin{lemma}[\cite{l2}] \label{lem2.1}
 $V$ is a Banach space equipped with the norm of \eqref{e2.1}.
\end{lemma}

\begin{proof}
 From the definition of $V$, we set $V=\{u+c: u\in W_0^{1,p}(\Omega),
 c\in\mathbb{R}\}$. We assume that $v_n\in V$, then $v_n=u_n+c_n$,
with $u_n\in W_0^{1,p}(\Omega)$. If $\{v_n\}$ is Cauchy sequence
in $W^{1,p}(\Omega)$, then for all $\varepsilon>0$, we have
\begin{align*}
\varepsilon>\|v_n-v_m\|_{W^{1,p}}
&= \|u_n+c_n-(u_m+c_m)\|_{W^{1,p}} \\
&= \|\nabla (u_n-u_m)\|_{L^{p}}+\|u_{n}-u_{m}+c_{n}-c_{m}\|_{L^{p}} \\
&\geq \|\nabla (u_n-u_m)\|_{L^{p}} .
\end{align*}
We obtain that $\{u_n\}$ is Cauchy sequence in $W_0^{1,p}(\Omega)$,
so there exists $\tilde{u}\in W_0^{1,p}(\Omega)$ such that
\[
u_n\to \tilde{u} \quad \text{in} W_0^{1,p}(\Omega).
\]
As
\[
\|u_n-u_m\|_{L^{p}}\leq c_{p}\|\nabla(u_n-u_m)\|_{L^{p}}\leq c_{p}\varepsilon,
\]
we have
\begin{align*}
\|c_n-c_m\|_{L^p}
&= \|u_n+c_n-(u_m+c_m)-u_n+u_m\|_{L^p} \\
&\leq \|u_n+c_n-(u_m+c_m)\|_{L^p}+\|u_n-u_m\|_{L^p} \\
&\leq \|v_n-v_m\|_{L^p}+c_{p}\|u_n-u_m\|_{L^p} \\
&\leq \varepsilon+c_{p}\varepsilon .
\end{align*}
We conclude that $\{c_n\}$ is a Cauchy sequence in $L^p(\Omega)$,
and so is in $\mathbb{R}$. We conclude that there
exists $\tilde{c}\in \mathbb{R}$, such that
\begin{equation*}
u_n+c_n\to \tilde{u}+\tilde{c} \quad
 \text{in $V$ as $c_n\to \tilde{c}$ in $\mathbb{R}$}.
\end{equation*}
\end{proof}

\begin{definition} \label{def2.2}\rm
 We say that $u\in V$ is a weak solution of
the boundary-value problem \eqref{e1.1}-\eqref{e1.3}if
\begin{equation}
\int_{\Omega}a(x,\nabla u)\cdot\nabla
v\,dx+\int_{\Omega}|u|^{p-2}uv\,dx
-\lambda\int_{\Omega}f(x,u)v\,dx=0, \quad \forall v \in V.
\end{equation}
\end{definition}

\begin{definition}[\cite{f1}] \label{def2.3}\rm
 Let $A:\Omega\times \mathbb{R}^{N}\to \mathbb{R}, A=A(x,\xi)$
be a continuous function in $\Omega\times\mathbb{R}^N$ with
continuous derivative with respect to $\xi$,
$a(x,\xi)=\nabla_{\xi} A(x,\xi)=A'$.
Define $A^{|\vee|}:\Omega\times\mathbb{R}\to\mathbb{R}$ as follows,
\[
A^{|\vee|}(x,t)=\sup_{|\xi|=t}A(x,\xi), \quad \forall x\in\Omega.
\]
For every $\varepsilon, b\in (0,1)$ and $x\in\Omega$, define
\begin{align*}
E_{\varepsilon,b}(x)
 =\Big\{&(\xi,\eta)\in\mathbb{R}^{N}\times\mathbb{R}^{N}:
A(x,\frac{\xi-\eta}{2})\geq\frac{1}{2}\max\{A(x,\varepsilon\xi),
A(x,\varepsilon\eta)\}, \\
&A(x,\frac{\xi+\eta}{2})>(1-b)\frac{A(x,\xi)+A(x,\eta)}{2}\Big\},
\end{align*}
and
\[
q_{\varepsilon,b}(x)=\sup\{\frac{|\xi-\eta|}{2}:
(\xi,\eta)\in E_{\varepsilon,b}(x)\}.
\]
We say that $A$ satisfies condition (UC) if
\[
\lim_{b\to0}\int_{\Omega}A^{|\vee|}(x,q_{\varepsilon,b}(x))dx=0 \quad
\text{for every } \varepsilon\in(0,1).
\]
So a function $A$ is said to be uniformly convex if $A$ satisfies
condition (UC).
\end{definition}

As in \cite{f1}, we remark that for $A(\xi)=|\xi|^{p}$,
 the $p$-uniform convexity condition
\[
A(x,\frac{\xi+\eta}{2})\leq\frac{1}{2}A(x,\xi)
+\frac{1}{2}A(x,\eta)-k|\xi-\eta|^{p}, \quad \forall
x\in\Omega,\; \xi,\eta\in \mathbb{R}^{N},
\]
where $k$ is a positive constant, is satisfied only if
$p\in[2,+\infty)$, but (UC) is satisfied for all $p\in(1,+\infty)$.

\begin{lemma}[\cite{k2,s1,z1}] \label{lem2.4}
 Let $X$ be a Banach space and $I\in C^{1}(X;\mathbb{R})$ satisfy
the Palais-Smale condition. Suppose
\begin{itemize}
\item[(i)]  $I(0)=0$;
\item[(ii)] there exists constants $r>0, a>0$ such that
$I(u)\geq a$ if $\|u\|=r$;
\item[(iii)] there exists $u_1\in X$ such that $\|u_1\|\geq r$
and $I(u_1)<a$.
\end{itemize}
Define
\[
\Gamma=\{\gamma\in C([0,1];X): \gamma(0)=0, \gamma(1)=u_1\}.
\]
Then
\[
\beta=\inf_{\gamma\in\Gamma}\sup_{u\in \gamma}I(u)\geq a
\]
is a critical value.
\end{lemma}

\section{Main result}

 Let $p>1$, $A:\Omega\times \mathbb{R}^{N}\to \mathbb{R}$, $a(x,\xi)$
 be derivative of $A(x,\xi)$ with respect to $\xi$, and we assume
that the following conditions hold
\begin{itemize}
\item[(A1)] $A(x,0)=0$ for all $x\in\Omega$;

\item[(A2)] $a$ satisfies the growth condition
$|a(x,\xi)|\leq c_{2}(1+|\xi|^{p-1})$ for all
$x\in\Omega$, $\xi\in \mathbb{R}^{N}$,
for some constant $c_{2}>0$;

\item[(A3)] $A$ is uniformly convex;

\item[(A4)] $A$ is $p$-subhomogeneous,
$0\leq a(x,\xi)\xi\leq p A(x,\xi)$  for all $x\in\Omega$,
$\xi\in \mathbb{R}^{N}$.

\item[(A5)] $A$ satisfies
$A(x,\xi)\geq \Lambda |\xi|^{p}$  for all $x\in\Omega$,
$\xi\in \mathbb{R}^{N}$,
where $\Lambda>0$ is a constant.
\end{itemize}
Let $f:\Omega\times \mathbb{R}\to \mathbb{R}$ be a continuous
function satisfying the following conditions:
\begin{itemize}
\item[(F1)] The subcritical growth condition
\[
|f(x,s)|\leq c_{3}(1+|s|^{q-1}), \quad\forall x\in\Omega,s\in
\mathbb{R},
\]
where $p<q<p^{\ast}=\frac{Np}{N-p}$ if $p<N$ or $p<q<+\infty$ if
$p>N$;

\item[(F2)] (The Ambrosetti-Rabinowitz condition)
$F(x,s)=\int_0^{s}f(x,t)dt$ is $\theta$-super\-homogeneous at
infinity; i.e., there exists $s_0>0$ such that
\[
0<\theta F(x,s)\leq f(x,s)s, \quad \text{for } |s|\geq s_0,\,
 x\in\Omega,
\]
where $\theta>p$;

\item[(F3)] $ \lim_{|s|\to0}\frac{f(x,s)}{|s|^{p-1}}=0$;

\item[(F4)] $ \lim_{|s|\to\infty}\frac{f(x,s)}{|s|^{p-1}}=0$;

\item[(F5)]  There exists $s^{\ast}>0, s^{\ast}\in \mathbb{R}$
such that $F(x,s^{\ast})>0$, $\forall x\in \Omega$.

\end{itemize}

Our main result is as follows.

\begin{theorem} \label{thm3.1}
 Let $A:\Omega\times \mathbb{R}^{N}\to \mathbb{R}^{N}$ be a
potential which satisfies {\rm (A1)--(A5)},
and let $f:\Omega\times \mathbb{R}\to \mathbb{R}$ be a continuous
function.
If $f$ satisfies {\rm (F1)--(F3)}, then \eqref{e1.1}-\eqref{e1.3}
has at least one nontrivial weak solution in $V$, for every
$\lambda\in \mathbb{R}$.
\end{theorem}

\begin{theorem} \label{thm3.2}
Let $A:\Omega\times \mathbb{R}^{N}\to \mathbb{R}^{N}$ be a potential
which satisfies {\rm (A1)--(A5)}, and let
$f:\Omega\times \mathbb{R}\to \mathbb{R}$ be a continuous function.
If $f$ satisfies {\rm (F3)--(F5)}, then there exists a constant
$\mu>0$, such that for $\lambda\in(\mu,+\infty)$,
 problem \eqref{e1.1}-\eqref{e1.3} has at least two nontrivial
weak solutions in $V$.
\end{theorem}

\subsection{Proof of Theorem \ref{thm3.1}}

Under the assumptions of Theorem \ref{thm3.1} we define the
functional
\[
J(u)=\int_{\Omega}A(x,\nabla u)\,dx
+\frac{1}{p}\int_{\Omega}|u|^{p}dx-\lambda\int_{\Omega}F(x,u)\,dx.
\]
It is easy to see that $J:V\to \mathbb{R}$ is
well defined and $J\in C^{1}(V;\mathbb{R})$. Its derivative is given
by
\[
\langle J'(u),\varphi\rangle=\int_{\Omega}a(x,\nabla
u)\cdot\nabla\varphi\, dx+\int_{\Omega}|u|^{p-2}u\varphi\, dx
-\lambda\int_{\Omega}f(x,u)\varphi\,dx,
\]
for all $u,\varphi\in V$. Thus the weak solution of
\eqref{e1.1}--\eqref{e1.3} corresponds to the critical point
of the functional $J$ on $V$.

To prove Theorem \ref{thm3.1}, we apply the mountain pass lemma to
this functional. We will show $J$ satisfies the Palais-Smale
condition in the first.
Let $\{u_{n}\}\subset V$ be a Palais-Smale sequence; i.e.,
$J'(u_{n})\to 0$ in $X^{\ast}$ and $J(u_{n})\to l$, where $l$
is a constant.
We first show that $\{u_{n}\}$ is bounded in $V$,
\begin{align*}
J(u_{n})-\frac{1}{\theta}\langle
J'(u_{n}),u_{n}\rangle
&= \int_{\Omega}[A(x,\nabla
u_n)-\frac{1}{\theta}a(x,\nabla u_n)\cdot\nabla u_{n}]dx\\
&\quad + (\frac{1}{p}-\frac{1}{\theta}) \int_{\Omega}|u_n|^{p}dx
+\lambda\int_{\Omega}[\frac{1}{\theta}f(x,u_{n})u_{n}-F(x,u_{n})]dx,
\end{align*}
where $\theta>p$. From condition (A4), we have
\begin{align*}
J(u_{n})-\frac{1}{\theta}\langle
J'(u_{n}),u_{n}\rangle
&\geq (1-\frac{p}{\theta})\int_{\Omega}A(x,\nabla
u_{n})dx+(\frac{1}{p}-\frac{1}{\theta})\int_{\Omega}|u_n|^{p}dx \\
&\quad + \lambda\int_{\Omega}[\frac{1}{\theta}f(x,u_{n})u_{n}-F(x,u_{n})]dx,
\end{align*}
then
\begin{align*}
&(1-\frac{p}{\theta})\int_{\Omega}A(x,\nabla u_n)dx
+ (\frac{1}{p}-\frac{1}{\theta})\int_{\Omega}|u_n|^{p}dx\\
&\leq J(u_{n})-\frac{1}{\theta}\langle J'(u_{n}),u_{n}\rangle
-\lambda\int_{\{x: |u_{n}(x)|>s_0\}}
 [\frac{1}{\theta}f(x,u_{n})u_{n}-F(x,u_{n})]dx+Mm(\Omega),
\end{align*}
where $M=\sup\{| \frac{1}{\theta}f(x,s)s-F(x,s)|:x\in\Omega, |s|\leq
s_0\}$, and $m(\Omega)$ denotes the Lebesgue measure of $\Omega$.

By  (F2) (the Ambrosetti-Rabinowitz condition), we have
\[
(1-\frac{p}{\theta})\int_{\Omega}A(x,\nabla u_n)dx
+(\frac{1}{p}-\frac{1}{\theta})\int_{\Omega}|u_n|^{p}dx\leq
J(u_{n})-\frac{1}{\theta}\langle
J'(u_{n}),u_{n}\rangle+Mm(\Omega).
\]
By  (A5),
\[ %(3.1)
(1-\frac{p}{\theta})\min\{\Lambda,\frac{1}{p}\}
(\int_{\Omega}|\nabla u_{n}|^{p}+|u_n|^{p})dx\leq
J(u_{n})-\frac{1}{\theta}\langle J'(u_{n}),u_{n}\rangle+Mm(\Omega),
\]
where $\min\{\Lambda,\frac{1}{p}\}$ denotes the minimum of
$\Lambda$ and $\frac{1}{p}$.
As
\[
\|u_n\|=\Big(\int_{\Omega}|\nabla
u_n|^{p}+|u_n|^{p}dx\Big)^{1/p},
\]
we conclude that $\{u_{n}\}$ is bounded in $V$.
Since $V$ is a closed subspace of $W^{1,p}(\Omega)$ and the
reflexivity of $W^{1,p}(\Omega)$, we may extract a weakly convergent
subsequence that we call $\{u_{n}\}$ for simplicity. So we may
assume that $u_{n}\rightharpoonup u$ weakly in $W^{1,p}(\Omega)$.

Next, we will prove
that ${u_{n}}$ converges strongly to $u\in V$.
From the derivative of $J$ we obtain
\begin{equation} \label{e3.2}
\begin{split}
&\int_{\Omega}a(x,\nabla
u_n)\cdot\nabla(u_n-u)dx+ \int_{\Omega}|u_n|^{p-2}u_n(u_n-u)dx \\
&= \langle J'(u_n),u_n-u\rangle-\lambda\int_{\Omega}f(x,u_n)(u_n-u)\, dx.
\end{split}
\end{equation}
Since $\|J'(u_n)\|_{W^{-1,p'}}\to0$ and $\{u_n-u\}$ is bounded
in $V\subset W^{1,p}(\Omega)$, by the
$|\langle J'(u_n),u_n-u\rangle|\leq\|J'(u_n)\|_{W^{-1,p'}}\|u_n-u\|$
it follows that
\[
\langle J'(u_n),u_n-u\rangle\to0.
\]
 From  (F1), we have
\begin{align*}
&\int_{\Omega}|f(x,u_n(x))||u_n(x)-u(x)|dx \\
&\leq c_3\int_{\Omega}|u_n(x)-u(x)|dx
+ c_3\int_{\Omega}|u_n(x)|^{q-1}|u_n(x)-u(x)|dx \\
&\leq c_3((m(\Omega))^{1/q'}+\|u_n\|_{L^{q}}^{q-1})
\|u_n-u\|_{L^q},
\end{align*}
where $\frac{1}{q}+\frac{1}{q'}=1$.
Since the embedding $W^{1,p}(\Omega)\hookrightarrow L^q(\Omega)$
is compact, with $q<\frac{Np}{N-p}$, we obtain $u_n\to u$ strongly
in $L^q(\Omega)$. So we obtain
\[
\int_{\Omega}|f(x,u_n(x))||u_n(x)-u(x)|dx\to0.
\]
Considering the inequality
\begin{align*}
\int_{\Omega}||u_n(x)|^{p-2}u_n(x)(u_n(x)-u(x))|dx
&= \int_{\Omega}|u_n(x)|^{p-1}|u_n(x)-u(x)|dx \\
&\leq \|u_n\|_{L^p}^{p-1}\|u_n-u\|_{L^p},
\end{align*}
and $u_n\to u$ strongly in $L^p(\Omega)$, we have
\[
\int_{\Omega}||u_n(x)|^{p-2}u_n(x)(u_n(x)-u(x))|dx .
\]
 From \eqref{e3.2}, we may conclude
\[ %3.3
\limsup_{n\to\infty}\langle a(x,u_{n}),u_{n}-u\rangle
=\limsup_{n\to\infty}\int_{\Omega}a(x,\nabla u_n)
 \cdot\nabla(u_n-u)dx\leq0,
\]
where $\langle a(x,u_{n}),u_{n}-u\rangle$ denotes
$\int_{\Omega}a(x,\nabla u_n)\cdot\nabla(u_n-u)dx$.

Therefore, from condition (A3), $A$ is uniformly convex, and
the operator $a(x,\xi)=D_{\xi}A(x,\xi)$ satisfies the $(S_+)$ property.
 From the $(S_+)$ condition in \cite[Proposition 2.1]{n1},
so we have $u_{n}\to u$ strongly in $W^{1,p}(\Omega)$. Since
$\{u_{n}\}\subset V$, $V$ is a closed subspace of $W^{1,p}(\Omega)$,
and we have $u\in V$. So $u_{n}\to u$ strongly in $V$.

Next, we show that $J$ satisfies the geometry condition of the
mountain pass lemma; i.e.,
\begin{itemize}
\item[(1)] There exists $r>0$, such that
$\inf_{\| u\|=r}J(u)=b>0$.

\item[(2)] There exists $u_0\in V$ such that
$J(tu_0)\to -\infty$, as $t\to+\infty$.
\end{itemize}

\noindent\textbf{Step 1.}
 Fix $\lambda\in \mathbb{R}$, we choose $\varepsilon>0$ small
enough satisfying
$\Lambda>\frac{\lambda\varepsilon}{pc_{p}}$.
Then by  (F3), there exists $\delta>0$ such that
$|f(x,s)|\leq\varepsilon|s|^{p-1}$ for $|s|\leq\delta$, for all
$x\in\Omega$. Integrating the above inequality, we deduce that
\[
F(x,s)\leq\frac{\varepsilon}{p}|s|^{p},\quad \text{for } |s|\leq\delta.
\]
Consequently, using  (F1) and the
Sobolev embedding, we have
\begin{align*}
 J(u)
&\geq \int_{\Omega}A(x,\nabla u)dx+\frac{1}{p}\int_{\Omega}|u|^{p}dx
 -\lambda\int_{\{x\in\Omega:|u(x)|\leq\delta\}}
 \frac{\varepsilon}{p}|u|^{p}dx\\
&\quad -\lambda\int_{\{x\in\Omega:|u|>\delta\}}c_{4} |u|^{q}dx\\
&\geq \min\{\Lambda,\frac{1}{p}\}\|u\|^{p}-\frac{\lambda\varepsilon}{p}c_{p}\|u\|^{p}-\lambda c_{4}\|u\|^{q} \\
&\geq \Big(\min\{\Lambda,\frac{1}{p}\}
-\frac{\lambda\varepsilon}{p}c_{p}\Big)\|u\|^{p}
-\lambda c_{4}\|u\|^{q}=\Phi(r),
\end{align*}
where $r=\|u\|^{p}$,
$\min\{\Lambda,\frac{1}{p}\}>\frac{\lambda\varepsilon}{p}c_p$, as
$\varepsilon$ is small enough. Moreover, $\Phi(r)>0$ for $r>0$
small enough, since $q>p$.

\noindent\textbf{Step 2.} Since $A$ is $p$-subhomogeneous,
can be restated as a differential inequality for the function $F$
in the form
\[
s|s|^{\theta}\frac{d}{ds}(|s|^{-\theta}F(x,s))\geq0, \quad
\text{for } |s|\geq s_0.
\]
We infer that for $|s|\geq s_0$, we have
$F(x,s)\geq\gamma_0(x)|s|^{\theta}$,
where
\[
\gamma_0=s_0^{-\theta}\min\{F(x,s_0),F(x,-s_0)\}>0.
\]
Considering  condition (A4), we obtain that for some constant
$k(u)>0$ there holds
\begin{align*}
 J(tu_0)
&= \int_{\Omega}A(x,t\nabla
u_0)dx+\frac{1}{p}\int_{\Omega}|tu_0|^{p}dx
-\lambda\int_{\Omega}F(x,tu_0)dx\\
&\leq t^{p}\int_{\Omega}A(x,\nabla
u_0)dx+\frac{1}{p}t^p\int_{\Omega}|u_0|^{p}dx-k(u)|\lambda|
t^{\theta}+|\lambda|M_{1}m(\Omega) ,
\end{align*}
where $M_{1}=\sup\{|F(x,s)|:x\in\Omega,|s|\leq s_0\}$. Since
$\theta>p$, we choose $u_0$ such that
$m\{x\in\Omega:u_0(x)\geq s_0\}>0$.
We deduce that $J(tu_0)\to-\infty$, as $t\to
+\infty$. For fixed $u_0\neq0$ and sufficiently large $t>0$, we
let $u_{1}=tu_0$.
By Lemma \ref{lem2.4} (mountain pass lemma), we obtain the
existence of a non-trivial solution $u$ to \eqref{e1.1}-\eqref{e1.3}.
The proof is completed.

\subsection{Proof of Theorem \ref{thm3.2}}

We denote
\[
\mathcal{A}(u)=\int_{\Omega}A(x,\nabla
u)dx+\frac{1}{p}\int_{\Omega}|u|^pdx
\]
and $\mathcal{F}(u)=\int_{\Omega}F(x,u)dx$, then the
functional $J$ is given by
$J(u)=\mathcal{A}(u)-\lambda\mathcal{F}(u)$.

\begin{lemma}[\cite{k1}] \label{lem3.3}
 For every $\lambda\in \mathbb{R}$, the functional
$J:V\to \mathbb{R}$ is sequentially weakly lower semicontinuous.
\end{lemma}

\begin{proof}
 The functional $\mathcal{A}$ being locally uniformly convex
is weakly lower semicontionous. From the condition $(F_4)$,
we have $|f(x,s)|\leq c_5(1+|s|^{p-1})$ for every $s\in \mathbb{R}$.
Since the embedding $V\subset W^{1,p}(\Omega)
\hookrightarrow L^{p}(\Omega)$ is compact, we obtain that
$\mathcal{F}$ is sequentially weakly lower semicontinuous
in the standard method.
\end{proof}

\begin{lemma} \label{lem3.4}
For every $\lambda\in \mathbb{R}$, the
functional $J$ is coercive and satisfies the Palais-Smale condition.
\end{lemma}

\begin{proof} By (F4), for $\varepsilon>0$ small enough,
there exists $\delta$ such that
$|f(x,s)|\leq\varepsilon|s|^{p-1}$ for every
$|s|\geq\delta$.
Integrating this inequality, we have
\[
|F(x,s)|\leq\frac{\varepsilon}{p}|s|^{p}
+\max_{|t|\leq\delta}|f(x,t)||s|,\quad \forall s\in \mathbb{R}.
\]
Thus, for every $u\in V$, we obtain
\begin{align*}
 J(u)
 &\geq \mathcal{A}(u)-|\lambda||\mathcal{F}(u)|\\
 &\geq \min\{\Lambda,\frac{1}{p}\}\|u\|^p
 -|\lambda|\frac{\varepsilon}{p}\int_{\Omega}|u|^{p}dx
 -|\lambda|\max_{|t|\leq\delta}|f(x,t)|\int_{\Omega}|u|dx\\
 &\geq \min\{\Lambda,\frac{1}{p}\}\|u\|^p
 -\frac{\varepsilon|\lambda|}{p}\int_{\Omega}|u|^{p}dx
 -|\lambda|m(\Omega)^{1/p'}\max_{|t|\leq\delta}|f(x,t)|
 \Big(\int_{\Omega}|u|^{p}dx\Big)^{1/p}\\
 &\geq \Big(\min\{\Lambda,\frac{1}{p}\}-\frac{\varepsilon|
\lambda|c_{p}}{p}\Big)\|u\|^p-c_{p}^{1/p}|
 \lambda|m(\Omega)^{1/p'}\max_{|t|\leq\delta}|f(x,t)|
\|u\|.
\end{align*}
Since $\varepsilon$ is small enough,
$\min\{\Lambda,\frac{1}{p}\}>\frac{\varepsilon|\lambda|c_{p}}{p}$,
so we have $J(u)\to+\infty$, whenever $\|u\|\to+\infty$.
Hence $J$ is coercive.

The proof of the functional $J$ satisfying the Palais-Smale
condition is similar to Theorem \ref{thm3.1}
The proof is complete.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm3.2}]
 From  condition (F5), we have
\begin{equation*}
\rho:=\sup_{u\in V,u\neq0}\frac{\mathcal{F}(u)}{\mathcal{A}(u)}
\geq\frac{\mathcal{F}(s^\ast)}{\mathcal{A}(s^\ast)}>0.
\end{equation*}
Let $\mu=1/\rho$. Fix $\lambda\in(\mu,+\infty)$.
From the definition of $\rho$, there exists some $u^\ast\in V$,
with $\min\{\mathcal{A}(u^\ast),\mathcal{F}(u^\ast)\}>0$, such that
\[
\frac{1}{\lambda}<\frac{\mathcal{F}(u^\ast)}{\mathcal{A}(u^\ast)}.
\]
This implies
$J(u^\ast)=\mathcal{A}(u^\ast)-\lambda\mathcal{F}(u^\ast)<0$.
By Lemma \ref{lem3.4}, the functional $J$ is bounded from below, coercive
and satisfies the (P-S) condition on $V$ for every $\lambda>0$.
This implies the functional $J$ has a global minimizer $u_1$; i.e.,
\[
J(u_1)\leq J(u) \quad  \forall u\in V.
\]
Let $u=u^\ast$. We have
\[
J(u_1)\leq J(u^\ast)<0.
\]
By  (F3), there exists $\delta>0$ such that
$|f(x,s)|\leq\varepsilon|s|^{p-1}$ for $|s|<\delta$, for all
$x\in \Omega$. We have
\begin{equation} \label{e3.5}
|F(x,s)|\leq\frac{\varepsilon}{p}|u|^p \text{for} |s|\leq\delta.
\end{equation}
Using  (F4), there exists $k(\delta)>0$ such that
$|F(x,s)|\leq k(\delta)|s|^p\leq k(\delta)|s|^q$,
$p<q<\frac{Np}{N-p}$, for $|s|>\delta$. Considering this fact
 and \eqref{e3.5}, for $\lambda\in(\mu,+\infty)$ we have
\begin{align*}
 J(u)
&\geq \int_{\Omega}A(x,\nabla
u)dx+\frac{1}{p}\int_{\Omega}|u|^{p}dx-\lambda\int_{\{x\in\Omega:|u(x)|\leq\delta\}}\frac{\varepsilon}{p}|u|^{p}dx\\
&\quad -\lambda\int_{\{x\in\Omega:|u|>\delta\}}k(\delta)
|u|^{q}dx\\
&\geq  \min\{\Lambda,\frac{1}{p}\}\|u\|^{p}-\frac{\lambda\varepsilon}{p}c_{p}\|u\|^{p}-\lambda k(\delta)\|u\|^{q} \\
&\geq \Big(\min\{\Lambda,\frac{1}{p}\}-\frac{\lambda\varepsilon}{p}c_{p}
\Big)\|u\|^{p}-\lambda k(\delta)\|u\|^{q}=\Phi(r),
\end{align*}
where $r=\|u\|^{p}$ and $q>p$. We can take $\varepsilon$ small enough,
such that $\min\{\Lambda,\frac{1}{p}\}>\frac{\lambda\varepsilon}{p}c_p$. Moreover, $\exists r>0$ small enough and $a>0$, such that $\Phi(r)\geq a>0$.\\
Obviously, $J(0)=0$. If we denote by $\Gamma$ the set of all
continuous functions $\gamma:[0,1]\to V$, such that
$\gamma(0)=0$ and $\gamma(1)=u_1$. From the mountain pass lemma,
there exists $u_2$ such that
$J'(u_2)=0$
and
\[
J(u_2)=\beta=\inf_{\gamma\in\Gamma}\sup_{u\in \gamma}J(u)\geq a>0.
\]
This completes the proof.
\end{proof}

\subsection*{Acknowledgements}
 The authors thank Professor V. Radulescu for his valuable
comments which helped to improve this article.

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\end{document}