\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 98, pp. 1--10.\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-2008/98\hfil Existence of solutions for a resonant problem]
{Existence of solutions for a resonant problem under
Landesman-Lazer conditions}

\author[Q. A. Ng\^o, H. Q. Toan\hfil EJDE-2008/98\hfilneg]
{Qu\^{o}\hspace{-0.5ex}\llap{\raise 1ex\hbox{\'{}}}\hspace{0.5ex}c 
Anh Ng\^o, Hoang Quoc Toan} % in alphabetical order

\address{Qu\^o\!$^{'}$\hspace{-0.5ex}c Anh Ng\^o \newline
Department of Mathematics\\
College of Science, Vi\^et Nam National University\\
H\`a N\^oi, Vi\^et Nam}
\email{bookworm\_vn@yahoo.com}
\email{nqanh@vnu.edu.vn}

\address{Hoang Quoc Toan \newline
Department of Mathematics \\
College of Science, Vi\^et Nam National University\\
H`a N\^oi, Vi\^et Nam}
\email{hq\_toan@yahoo.com}

\thanks{Submitted March 24, 2008. Published July 25, 2008.}
\subjclass[2000]{35J20, 35J60, 58E05}
\keywords{$p$-Laplacian; Non-uniform; Landesman-Laser type;
Divergence form}

\begin{abstract}
This article shows the existence of weak solutions in $W_0^1
(\Omega )$ to a class of Dirichlet problems of the form
\[
- \operatorname{div}({a({x,\nabla u} )})= \lambda_1 |u|^{p - 2} u
+ f(x,u)-h
\]
in a bounded domain $\Omega$ of $\mathbb{R}^N$. Here $a$ satisfies
\[
|{a({x,\xi } )}| \leq c_0 \big({h_0 (x)+ h_1 (x )|\xi|^{p - 1}}\big)
\]
for all $\xi \in \mathbb{R}^N$, a.e. $x \in \Omega$, $h_0 \in
L^{\frac{p}{p - 1}} (\Omega )$, $h_1 \in L_{\rm loc}^1 ( \Omega )$,
$h_1(x) \geq 1$ for a.e. $x$ in $\Omega$; $\lambda_1$ is the first
eigenvalue for $-\Delta_p$ on $\Omega$ with zero Dirichlet
boundary condition and $g$, $h$ satisfy some suitable conditions.
\end{abstract}

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

\section{Introduction}

Let $\Omega$ be a bounded domain in $\mathbb{R}^N$. In the present paper we study the existence of weak solutions of the following Dirichlet problem
\begin{equation}\label{btbd}
-\operatorname{div}({a({x,\nabla u} )})= \lambda_1 |u|^{p - 2} u
+ f(x,u)-h
\end{equation}
where $|{a({x,\xi } )}| \leq c_0 ({h_0 ( x )+ h_1 (x )|\xi |^{p -
1} } )$ for any $\xi$ in $\mathbb{R}^N$ and a.e. $x \in \Omega$,
$h_0(x) \geq 0$ and $h_1(x) \geq 1$ for any $x$ in $\Omega$.
$\lambda_1$ is the first eigenvalue for $-\Delta_p$ on $\Omega$
with zero Dirichlet boundary condition. We define $X := W_0^{1,p}
( \Omega )$ as the closure of $C_0^\infty(\Omega)$ under the norm
$$
\| u \| = \Big({\int_\Omega {|{\nabla u}|^p \,dx} } \Big)^{1/p}.
$$
It is well-known that
\[
\lambda _1 = \inf_{u \in W_0^{1,p} (\Omega )}
\bigg\{ {\int_\Omega {|{\nabla u}|^p \,dx} : {\int_\Omega {|u |^p
\,dx} = 1} } \bigg\}.
\]
Recall that $\lambda_1$ is simple and positive. Moreover, there
exists a unique positive eigenfunction $\phi_1$ whose norm in
$W_0^{1,p}(\Omega)$ equals to one. Regarding the functions $f$, we
assume that $f$ is a bounded Carath\'{e}odory function. We also
assume that $h \in L^{p'}(\Omega )$ where $p' = \frac{p}{p - 1}$.

In the present paper, we study the case in which $h_0$ and $h_1$
belong to $L^{\frac{p}{p - 1}} (\Omega )$ and $L_{\rm loc}^1 (\Omega
)$, respectively. The problem now may be non-uniform in sense that
the functional associated to the problem may be infinity for some
$u$ in $X$. Hence, weak solutions of the problem must be found in
some suitable subspace of $X$. To our knowledge, such problems
were firstly studied by \cite{DV, V, TN}. In order to state our
main theorem, let us introduce our hypotheses on the structure of
problem \eqref{btbd}.

Assume that $N \geq 1$ and $p > 1$. $\Omega$ be a bounded domain
in $\mathbb{R}^N$ having $C^2$ boundary $\partial \Omega$. Consider
$a:\mathbb{R}^N \times \mathbb{R}^N \to \mathbb{R}^N$, $a = a({x,\xi
} )$, as the continuous derivative with respect to $\xi$ of the
continuous function
$A: \mathbb{R}^N \times \mathbb{R}^N \to \mathbb{R}$,
$A = A({x,\xi } )$, that is, $a({x,\xi }) = \frac{\partial
A({x,\xi } )}{\partial \xi }$. Assume that there are a positive
real number $c_0$ and two nonnegative measurable functions $h_0$,
$h_1$ on $\Omega$ such that $h_1 \in L_{\rm loc}^1 (\Omega )$, $h_0
\in L^{\frac{p}{p-1}} ( \Omega )$, $h_1(x) \geq 1$ for a.e. $x$ in
$\Omega$.

Suppose that $a$ and $A$ satisfy the hypotheses below
\begin{itemize}
 \item[(A1)] $|{a({x,\xi } )}| \leq c_0 ({h_0 (x)+ h_1 (x )|\xi|^p-1 } )$
 for all $\xi \in \mathbb{R}^N$, a.e. $x \in \Omega$.

 \item[(A2)] There exists a constant $k_1 >0$ such that
 \[
A\Big({x,\frac{\xi + \psi }{2}}\Big)\leq \frac{1}{2}A( {x,\xi })+
\frac{1}{2}A({x,\psi })- k_1 h_1 (x )|{\xi - \psi }|^p
\]
for all $x$, $\xi$, $\psi$, that is, $A$ is $p$-uniformly convex

\item[(A3)] $A$ is $p$-subhomogeneous, that is,
\[
0 \leq a({x,\xi } )\xi \leq pA({x,\xi })
\]
for all $\xi \in \mathbb{R}^N$, a.e. $x \in \Omega$.

 \item[(A4)] There exists a constant $k_0 \geq 1/p$ such that
\[
A({x,\xi })\geq k_0 h_1 (x )|\xi|^p
\]
for all $\xi \in \mathbb{R}^N$, a.e. $x \in \Omega$.

\item[(A5)] $A(x,0)=0$ for all $x \in \Omega$.

\end{itemize}
A special case of \eqref{btbd} is the following equation involving
the $p$-Laplacian operator
\begin{equation}\label{btbd'}
- \operatorname{div}(|\nabla u|^{p-2} \nabla u)= \lambda_1 | u
|^{p - 2} u + f(x,u)-h
\end{equation}
We refer the reader to \cite{DV, M, NM, TN, V} for more examples.
We suppose also that there exists
\begin{itemize}
 \item[(H1)]
\[
\lim_{x \to - \infty } f({x,s})= f_{ - \infty }
(x)\quad , \quad\lim_{x \to + \infty } f({x,s}
)= f^{ + \infty } (x)
\]
for almost every $x \in \Omega$.
\end{itemize}
As is well-known, under (H1), problem \eqref{btbd'} may not have solutions. In \cite{AG, BDK, AO}, the existence of solutions of \eqref{btbd'} is shown provided that one of the following two conditions are satisfied
\begin{itemize}

\item[(H2)]
\[
\int_\Omega {f^{ + \infty } (x )\phi _1 (x )\,dx} < \int_\Omega
{h(x )\phi _1 (x )\,dx} < \int_\Omega {f_{ - \infty } (x )\phi _1
(x )\,dx} .
\]

\item[(H2')]
\[
\int_\Omega {f_{ - \infty } (x )\phi _1 (x )\,dx} < \int_\Omega
{h(x )\phi _1 (x )\,dx} < \int_\Omega {f^{ + \infty } (x )\phi _1
(x )\,dx} .
\]

\end{itemize}
It should be noticed that though conditions (H2) and (H2') look rather
similar, the existence proof in \cite{AO} is different in each case.
Indeed, under (H2) the functional associated with the problem is
coercive and achieves a minimum, whereas under (H2') the functional
has the geometry of the saddle point theorem. In the present paper,
we only consider the case (H2), the case (H2') is still an open
question due to the fact that there are some difficulties in
verifying geometric conditions of the saddle point theorem.

We also point out that in that papers, the property  $pA(x, \xi) =
a(x, \xi) \cdot \xi$, which may not hold under our assumptions by
(A4), play an important role in the arguments. In this paper, we
shall extend some results in \cite{AG, BDK, AO} in two directions:
one is from $p$-Laplacian operators to general elliptic operators
in divergence form and the other is to the case on non-uniform
problem.

Since problem \eqref{btbd} may be non-uniform, then we must
consider the problem in a suitable subspace of $X$. In fact, we
consider the following subspace of $W_0^{1,p} (\Omega )$
\begin{equation}
E = \bigg\{ {u \in W_0^{1,p} (\Omega ):\int_\Omega {h_1 (x )|
{\nabla u}|^p \,dx} < + \infty } \bigg\}.
\end{equation}
The space $E$ can be endowed with the norm $\| u \|_E =
({\int_\Omega {h_1 (x )|{\nabla u}|^p \,dx} } )^{1/p}$. As
in \cite{DV}, it is known that $E$ is an infinite dimensional
Banach space. We say that $u \in E$ is a weak solution for problem
\eqref{btbd} if
\[
\int_\Omega {a({x,\nabla u} )\nabla \phi \,dx} - \lambda_1
\int_\Omega {|u|^{p - 2} u\phi \,dx} - \int_\Omega {f(x,u )\phi
\,dx} + \int_\Omega {h\phi \,dx} = 0
\]
for all $\phi \in E$. Letting
\begin{gather*}
F({x,t})= \int_0^t {f({x,s})\,ds} , \\
J(u)= \frac{\lambda_1}{p} \int_\Omega {|u|^p \,dx} + \int_\Omega
{F(x,u )\,dx} - \int_\Omega {h u \,dx} , \\
\Lambda (u)= \int_\Omega {A({x,\nabla u} )\,dx}, \\
I(u)= \Lambda (u)- J(u)
\end{gather*}
for all $u \in E$. The following remark plays an important role
in our arguments.

\begin{remark} \label{chuy}\rm \quad
\begin{itemize}
 \item[(i)] $\| u \| \leq \| u \|_E$ for all $u \in E$ since $h_1(x) \geq 1$.

 \item[(ii)] By (A1), $A$ satisfies the growth condition
\[
|{A({x,\xi } )}| \leq c_0 ({h_0 (x )|\xi| + h_1 (x )|\xi |^p } )
\]
for all $\xi \in \mathbb{R}^N$, a.e. $x \in \Omega$.

 \item[(iii)] By (ii) above and (A4), it is easy to see that
\[
E = \big\{ {u \in W_0^{1,p} (\Omega ):\Lambda (u)< + \infty }\big\}
= \big\{ {u \in W_0^{1,p} (\Omega ):I(u) < + \infty }\big\}.
\]

 \item[(iv)] $C_0^\infty (\Omega)\subset E$ since $|{\nabla u}|$
 is in $C_c (\Omega )$ for any $u \in C_0^\infty (\Omega )$
 and $h_1 \in L_{\rm loc}^1 (\Omega )$.

 \item[(v)] By (A4) and Poincar\'{e} inequality, we see that
\[
\int_\Omega {A({x,\nabla u } )\,dx} \geq \frac{1} {p}\int_\Omega
{|{\nabla u }|^p \,dx} \geq \frac{\lambda _1 }{p}\int_\Omega {|{u
}|^p \,dx} ,
\]
for all $u\in W_0^{1,p} (\Omega )$.
\end{itemize}
\end{remark}
Now we describe our main result.

\begin{theorem}\label{dlc1}
Assume conditions {\rm (A1)--(A5), (H1)--(H2)} are
fulfilled. Then problem \eqref{btbd} has at least a weak solution
in $E$.
\end{theorem}

\section{Auxiliary results}

Due to the presence of $h_1$, the functional $\Lambda$ may not
belong to $C^1(E, \mathbb{R})$. This means that we cannot apply
the Minimum Principle directly. In this situation, we need some
modifications.

\begin{definition} \label{def1} \rm
Let $\mathcal{F}$ be a map from a Banach space $Y$ to $\mathbb{R}$.
We say that $\mathcal{F}$ is weakly continuous differentiable on $Y$
if and only if following two conditions are satisfied
\begin{itemize}
 \item[(i)] For any $u \in Y$ there exists a linear map $D\mathcal{F}(u)$ from $Y$ to $\mathbb{R}$ such that
\[
\lim_{t \to 0} \frac{\mathcal{F}({u + tv}) -
\mathcal{F}(u )}{t} = \langle D\mathcal{F}(u ), v \rangle
\]
for every $v \in Y$.
 \item[(ii)] For any $v \in Y$, the map
$u \mapsto \langle D\mathcal{F}(u ), v \rangle$ is continuous on $Y$.
\end{itemize}
\end{definition}

Denote by $C_w^1(Y)$ the set of weakly continuously differentiable
functionals on $Y$. It is clear that $C^1(Y) \subset C_w^1(Y)$
where we denote by $C^1(Y)$ the set of all continuously
Fr\'{e}chet differentiable functionals on $Y$.
Now let $\mathcal{F} \in C_w^1(Y)$, we put
\[
\| {D\mathcal{F}(u )} \| = \sup \{ | \langle
D\mathcal{F}(u ), h \rangle : |{h \in Y,\| h \| = 1}   \}
\]
for any $u \in Y$, where $\| {D\mathcal{F}(u )} \| $ may
be $+\infty$.

\begin{definition} \label{def2} \rm
We say that $\mathcal{F}$ satisfies the Palais-Smale condition if
any sequence $\{u_n\} \subset Y$ for which $\mathcal{F}(u_n)$
is bounded and $\lim_{n \to \infty } \| {D\mathcal{F}(u_n)} \| = 0$
possesses a convergent subsequence.
\end{definition}

The following theorem is our main ingredient.

\begin{theorem}[The Minimum Principle, see \cite{NT}]\label{bd:tmp}
Let $\mathcal{F} \in C_w^1(Y)$ where $Y$ is a Banach space. Assume that
\begin{itemize}
  \item[(i)]  $\mathcal{F}$ is bounded from below, $c = \inf \mathcal{F}$,
  \item[(ii)] $\mathcal{F}$ satisfies Palais-Smale condition.
\end{itemize}
Then there exists $u_0 \in Y$ such that $\mathcal{F}(u_0)=c$.
\end{theorem}

The proof of Theorem \ref{bd:tmp} is similar to the proof of
Theorem 3.1 in \cite{C} where we need a modified Deformation Lemma
which is proved in \cite[Theorem 2.2]{V}. For simplicity of notation,
we shall denote $D\mathcal{F}(u)$ by $\mathcal{F}'(u)$.
The following lemma concerns the smoothness of the functional $\Lambda$.

\begin{lemma}[\cite{DV}]\label{bd:tcLambda}\quad
\begin{itemize}
 \item[(i)] If $\{{u_n }\}$ is a sequence weakly converging to $u$ in $X$, denoted by $u_n \rightharpoonup u$,
then $\Lambda(u)\leq \liminf_{n \to \infty }
\Lambda(u_n)$.
 \item[(ii)] For all $u,z \in E$
\[
\Lambda \bigg({\frac{u + z}{2}}\bigg)\leq \frac{1}{2}\Lambda ( u)+
\frac{1}{2}\Lambda (z)- k_1 \| {u - z} \|_E^p .
\]

\item[(iii)] $\Lambda$ is continuous on $E$.

\item[(iv)] $\Lambda$ is weakly continuously differentiable on $E$ and
\[
\langle {\Lambda '(u ),v} \rangle = \int_\Omega
{a({x,\nabla u} )\nabla v\,dx}
\]
for all $u,v \in E$.

\item[(v)] $\Lambda (u)- \Lambda (v)\geq \langle {\Lambda '(v ),u - v} \rangle$ for all $u,v \in E$.
\end{itemize}
\end{lemma}

The following lemma concerns the smoothness of the functional $J$.
The proof is standard and simple, so we omit it.

\begin{lemma}\label{bd:tcJ}\quad
\begin{itemize}
 \item[(i)] If $u_n \rightharpoonup u$ in $X$, then $\lim_{n \to \infty } J(u_n) = J(u )$.
 \item[(ii)] $J$ is continuous on $E$.
 \item[(iii)] $J$ is weakly continuously differentiable on $E$ and
\[
\langle {J'(u ),v} \rangle = \lambda_1 \int_\Omega {|u
|^{p - 2} uv\,dx} + \int_\Omega {f({x,u} )v\,dx} - \int_\Omega
{hv\,dx}
\]
for all $u,v \in E$.
\end{itemize}
\end{lemma}

\section{Proofs}

We remark that the critical points of the functional $I$ correspond to
the weak solutions of \eqref{btbd}. Throughout this paper,
we sometimes denote by "const" a positive constant.

\begin{lemma}\label{bd:tcI}
The functional $I$ satisfies the Palais-Smale condition on $E$ provided
{\rm (H2)} holds.
\end{lemma}

\begin{proof}
Let $\{ {u_n }\}$ be a sequence in $E$ and $\beta$ be a real number such that
\begin{equation}\label{dkps1}
| I(u_n)| \leq \beta \quad {\text{for all }} n
\end{equation}
and
\begin{equation}\label{dkps2}
I'(u_n) \to 0 \quad{\text{in }} E^\star.
\end{equation}
We prove that $\{{u_n }\}$ is bounded in $E$.
We assume by contradiction that $\| {u_n } \|_E \to \infty $
as $n \to \infty $.

Let $v_n = u_n /\| {u_n } \|_E$ for every $n$.
Thus $\{{v_n }\}$ is bounded in $E$. By Remark \ref{chuy}(i),
we deduce that $\{{v_n }\}$ is bounded in $X$. Since $X$ is reflexive,
then by passing to a subsequence, still denoted by $\{{v_n }\}$,
we can assume that the sequence $\{{v_n }\}$ converges weakly to
some $v$ in $X$. Since the embedding $X \hookrightarrow L^p(\Omega)$
is compact then $\{{v_n }\}$ converges strongly to $v$ in $L^p(\Omega)$.

Dividing \eqref{dkps1} by $\| {u_n } \|_E ^p$ together with
Remark \ref{chuy}(v), we deduce that
\[
\limsup_{n \to + \infty }
\Bigg({\frac{1}{p}\int_\Omega {|{\nabla v_n }|^p \,dx} -
\frac{\lambda_1}{p} \int_\Omega {|{v_n }|^p \,dx} - \int_\Omega
{\frac{F({x,u_n } )}{\| {u_n } \|_E ^p }\,dx} +
\int_\Omega {h\frac{u_n }{\| {u_n } \|_E ^p }\,dx}
}\Bigg)\leq 0.
\]
Since, by the hypotheses on $p$, $f$, $h$ and $\{u_n\}$,
\[
\limsup_{n \to + \infty } \Bigg({\int_\Omega
{\frac{F({x,u_n } )}{\| {u_n } \|_E ^p }\,dx} +
\int_\Omega {h\frac{u_n }{\| {u_n } \|_E ^p }\,dx} } \Bigg)=
0,
\]
while
\[
\limsup_{n \to + \infty } \int_\Omega {|{v_n }
|^p \,dx} = \int_\Omega {|v|^p \,dx} ,
\]
we have
\[
\limsup_{n \to + \infty } \int_\Omega
{|{\nabla v_n }|^p \,dx} \leq \lambda _1 \int_\Omega {|v|^p \,dx} .
\]
Using the weak lower semi-continuity of norm and Poincar\'{e} inequality,
we get
\begin{align*}
 \lambda _1 \int_\Omega {|v|^p \,dx}
&\leq \int_\Omega {|{\nabla v}|^p \,dx}
 \leq \liminf_{n \to + \infty } \int_\Omega {|{\nabla v_n }|^p \,dx} \hfill \\
&\leq \limsup_{n \to + \infty } \int_\Omega {|{\nabla v_n }|^p \,dx}
 \leq \lambda _1 \int_\Omega {|v|^p \,dx}.
\end{align*}
Thus, these inequalities are indeed equalities. Besides, $\{{v_n }\}$ converges strongly to $v$ in $X$ and
\[
\int_\Omega {|{\nabla v}|^p \,dx} = \lambda _1 \int_\Omega {|v |^p
\,dx} .
\]
This implies, by the definition of $\phi_1$, that $v = \pm \phi_1$.

On the other hand, by means of \eqref{dkps1}, we deduce that
\begin{equation}\label{eq6}
\begin{split}
 - \beta p
&\leq -p \int_\Omega {A({x,\nabla u_n } )\,dx}
 + \lambda_1 \int_\Omega {|{u_n }|^p \,dx}
 + p \int_\Omega {F({x,u_n } )\,dx} \\
& \quad -p \int_\Omega {hu_n \,dx} \\
&\leq \beta p.
\end{split}
\end{equation}
In view of \eqref{dkps2}, there exists a sequence of positive real
numbers $\{ \varepsilon_n \}_n$ such that $\varepsilon_n \to 0$ as
$n \to +\infty$ and
\begin{equation}\label{eq7}
\begin{split}
 - \varepsilon _n \| {u_n } \|_E
&\leq \int_\Omega {a({x,\nabla u_n } )\nabla u_n\,dx}
 - \lambda_1 \int_\Omega {|{u_n }|^p \,dx}
 - \int_\Omega {f({x,u_n } )u_n \,dx} +\\
& \quad  \int_\Omega {hu_n \,dx} \\
& \leq \varepsilon _n \| {u_n } \|_E .
\end{split}
\end{equation}
Letting
\[
g({x,s})= \begin{cases}
 \dfrac{F({x,s} )}{s} & {\text{if }} s \ne 0,  \\
 f({x,0})&{\text{if }}s = 0.
\end{cases}
\]
We then consider the following two cases.

\noindent {\bf Case 1}: Suppose that $v_n \to -\phi_1$. Letting
$n \to +\infty$. Since $u_n(x) \to -\infty$, it follows that
\begin{gather*}
f({x,u_n (x )})\to f_{ - \infty } (x ), {\text{ a.e }} x \in
\Omega,\\
g({x,u_n (x )})\to f_{ - \infty } (x ), {\text{ a.e }} x \in
\Omega .
\end{gather*}
Therefore, the properties of $f$ and $F$, the Lebesgue theorem then imply
\begin{equation}\label{eq8}
\lim_{n \to + \infty } \int_\Omega {({f( {x,u_n
} )v_n - pg({x,u_n } )v_n } )\,dx} = (p-1 )\int_\Omega {f_{ -
\infty } (x )\phi _1 (x )\,dx} .
\end{equation}
On the other hand, by summing up \eqref{eq6} and \eqref{eq7}, we get
\begin{align*}
\int_\Omega ( pF( {x,u_n } ) &- f( {x,u_n } )u_n  )\,dx + ( {1 - p} )\int_\Omega {hu_n \,dx} \\
&\geq  \int_\Omega {( {a( {x,\nabla u_n } )\nabla u_n
 - pA( {x,\nabla u_n } )} )\,dx}+ \\
& \quad  \int_\Omega {( {pF( {x,u_n } ) - f( {x,u_n } )u_n } )\,dx} + ( {1 - p} )\int_\Omega {hu_n \,dx} \\
&\geq  - \beta p - \varepsilon _n \| {u_n } \|_E,
\end{align*}
and after dividing by $\|u_n\|_E $, we obtain
\begin{equation}\label{eq9}
\int_\Omega {({pg({x,u_n })- f({x,u_n } )v_n } )\,dx} + ({1 - p}
)\int_\Omega {hv_n \,dx}  \geq - \frac{\beta p}{\| {u_n }
\|_E } - \varepsilon _n .
\end{equation}
Since $h \in L^{p'}$ and $\| v_n - (-\phi_1) \|_X \to 0$, we obtain
\begin{equation}\label{eq10}
\lim_{n \to + \infty } \int_\Omega  {h v_n \,dx}
 =  -\int_\Omega  {h\phi _1 \,dx}.
\end{equation}
In \eqref{eq9}, taking $\liminf$ to both sides together with \eqref{eq8}
and \eqref{eq10}, we  deduce
\[
({1 - p} )\int_\Omega  {f_{ - \infty } (x )\phi _1 (x )\,dx} - ({1
- p} )\int_\Omega {h\phi _1 \,dx} \geq 0,
\]
which gives
\[
(p-1 )\int_\Omega {h\phi _1 (x )\,dx} \geq (p-1 )\int_\Omega {f_{ -
\infty } (x )\phi _1 (x )\,dx}
\]
which yields, since $p > 1$,
\[
\int_\Omega {h\phi _1 (x )\,dx} \geq \int_\Omega {f_{ - \infty } (x
)\phi _1 (x )\,dx}
\]
which contradicts (H2).
\smallskip

\noindent {\bf Case 2}: Suppose that $v_n \to \phi_1$. Letting $n \to +\infty$.
Since $u_n(x) \to \infty$,
\begin{gather*}
f({x,u_n (x )})\to f^{+ \infty } (x ), {\text{ a.e }} x \in \Omega, \\
g({x,u_n (x )})\to f^{+ \infty } (x ), {\text{ a.e }} x \in \Omega.
\end{gather*}
Therefore, by Lebesgue theorem,
\begin{equation}\label{eq11}
\lim_{n \to + \infty } \int_\Omega {({f( {x,u_n
} )v_n - pg({x,u_n } )v_n } )\,dx} = ({1-p} )\int_\Omega {f^{ +
\infty } (x )\phi _1 (x )\,dx} .
\end{equation}
On the other hand, by summing up \eqref{eq6} and \eqref{eq7}, we get
\begin{align*}
\int_\Omega (pF({x,u_n })&- f({x,u_n } )u_n  )\,dx  +  (p-1 )\int_\Omega {hu_n \,dx} \\
&\leq  \int_\Omega {({pA({x,\nabla u_n })- a({x,\nabla u_n } )\nabla u_n} )\,dx} + \\
& \quad  \int_\Omega {\big({pF({x,u_n })- f({x,u_n } )u_n } \big)\,dx} +  (p-1 )\int_\Omega {hu_n \,dx}\\
&\leq  \beta p + \varepsilon _n \| {u_n } \|_E,
\end{align*}
and after dividing by $\|u_n\|_E $, we obtain
\begin{equation}\label{eq12}
- \int_\Omega {({pg({x,u_n })- f({x,u_n } )v_n } )\,dx} + (p-1
)\int_\Omega {hv_n \,dx} \leq - \frac{\beta p}{\| {u_n }
\|_E } - \varepsilon _n.
\end{equation}
Since $h \in L^{p'}$ and $\| v_n - \phi_1 \|_X \to 0$,
\begin{equation}\label{eq13}
\lim_{n \to + \infty } \int_\Omega  {h v_n \,dx} =  \int_\Omega  {h\phi _1 \,dx}.
\end{equation}
In \eqref{eq12}, using \eqref{eq11}, \eqref{eq13} and by
taking $\limsup$ to both sides, we  deduce
\[
(p-1 )\int_\Omega {h\phi _1 (x )\,dx} \leq (p-1 )\int_\Omega {f^{+
\infty } (x )\phi _1 (x )\,dx}
\]
which yields, since $p > 1$,
\[
\int_\Omega {h\phi _1 (x )\,dx} \leq \int_\Omega {f^{ + \infty } (x
)\phi _1 (x )\,dx}
\]
which contradicts (H2).

From the two cases above, $\{{u_n }\}$ is bounded in $E$.
By Remark \ref{chuy}(i),  we deduce that $\{{u_n }\}$ is bounded in $X$.
Since $X$ is reflexive, then by passing to a subsequence, still
denote by $\{{u_n }\}$, we can assume that the sequence $\{{u_n }\}$
 converges weakly to some $u$ in $X$. We shall prove that the sequence
$\{{u_n }\}$ converges strongly to $u$ in $E$.

We observe by Remark \ref{chuy}(iii) that $u \in E$. Hence
$\{{\| {u_n - u} \|_E }\}$ is bounded.
Since $\{{\| {I' ({u_n - u} )} \|_{E^\star} }\}$
converges to 0, then ${\langle {I'({u_n - u} ),u_n - u} \rangle }$
converges to 0.

By the hypotheses on $f$ and $h$, we deduce that
\begin{gather*}
 \lim_{n \to + \infty } \int_\Omega {|{u_n }|^{p - 2} u_n ({u_n - u} )\,dx}
= 0, \\
\lim_{n \to + \infty } \int_\Omega {f({x,u_n } )({u_n - u} )\,dx} = 0, \\
\lim_{n \to + \infty } \int_\Omega {h({u_n - u} )\,dx} = 0.
\end{gather*}
On the other hand,
\begin{align*}
&\langle {J'(u_n),u_n - u} \rangle \\
&= \lambda_1 \int_\Omega {|{u_n }|^{p - 2} u_n ({u_n - u} )\,dx}
 + \int_\Omega {f({x,u_n } )({u_n - u} )\,dx}
 + \int_\Omega {h({u_n - u} )\,dx} .
\end{align*}
Thus
$\lim_{n \to \infty } \langle {J'(u_n),u_n - u} \rangle = 0$.
This and the fact that
\[
\langle {\Lambda '(u_n),u_n - u} \rangle = \langle {I'(u_n),u_n - u} \rangle
+ \langle {J'(u_n),u_n - u} \rangle
\]
give
\[
\lim_{n \to \infty } \langle {\Lambda'(u_n),u_n - u} \rangle = 0.
\]
By using (v) in Lemma \ref{bd:tcLambda}, we get
\[
\Lambda (u)- \mathop {\limsup }_{n \to \infty } \Lambda
(u_n) = \mathop {\liminf }_{n \to \infty } ( {\Lambda (u )-
\Lambda (u_n)})\geq \lim_{n \to \infty }
\langle {\Lambda '(u_n),u - u_n } \rangle = 0.
\]
This and (i) in Lemma \ref{bd:tcLambda} give
\[
\lim_{n \to \infty } \Lambda (u_n) = \Lambda (u).
\]
Now if we assume by contradiction that $\| {u_n - u} \|_E$ does not
converge to 0 then there exists $\varepsilon >0$ and a subsequence
$\{{u_{n_m } }\}$ of $\{{u_n }\}$ such that
\[
\| {u_{n_m } - u} \|_E \geq \varepsilon .
\]
By using relation (ii) in Lemma \ref{bd:tcLambda}, we get
\[
\frac{1}{2}\Lambda (u)+ \frac{1}{2}\Lambda \big({u_{n_m } }\big)
- \Lambda \bigg({\frac{u_{n_m } + u}{2}} \bigg)\geq k_1 \| {u_{n_m } - u}
\|_E^p \geq k_1 \varepsilon ^p .
\]
Letting $m \to \infty$ we find that
\[
\limsup_{m \to \infty } \Lambda \bigg({\frac{u_{n_m
} + u}{2}}\bigg)\leq \Lambda (u)- k_1 \varepsilon ^p .
\]
We also have $\dfrac{u_{n_m } + u}{2}$ converges weakly to $u$ in $E$.
Using (i) in Lemma \ref{bd:tcLambda} again, we get
\[
\Lambda (u)\leq \liminf_{m \to \infty }
\Lambda \bigg({\frac{u_{n_m } + u}{2}}\bigg).
\]
That is a contradiction. Therefore $\{{u_n }\}$ converges strongly
to $u$ in $E$.
\end{proof}

\begin{lemma}\label{bd:tcI_coercive}
The functional $I$ is coercive on $E$ provided {\rm (H2)} holds.
\end{lemma}

\begin{proof}
We firstly note that, in the proof of the Palais-Smale condition,
we have proved that if $I(u_n)$ is a sequence bounded from above
with $\|u_n\|_E \to \infty$, then (up to a subsequence),
$v_n = u_n/\|u_n\|_E \to \pm \phi_1$ in $X$. Using this fact,
we will prove that $I$ is coercive provided $(\bf H_3)$ holds.

Indeed, if $I$ is not coercive, it is possible to choose a sequence
$\{u_n\} \subset E$ such that $\|u_n\|_E  \to \infty$,
$I(u_n) \leq \operatorname{const}$ and
$v_n = u_n/\|u_n\|_E  \to \pm \phi_1$ in $X$.
By Remark \ref{chuy} (v),
\begin{equation}\label{(8)}
 - \int_\Omega {F({x,u_n } )\,dx} + \int_\Omega {hu_n \,dx} \leq I(u_n).
\end{equation}

\noindent {\bf Case 1}: Assume that $v_n \to \phi_1$.
Dividing \eqref{(8)} by $\|u_n\|_E $ we get
\begin{align*}
 - \int_\Omega {f^{ + \infty } \phi _1 \,dx}
 + \int_\Omega {h\phi _1 \,dx}
&= \lim_{n \to + \infty }
 \Bigg({ - \int_\Omega {\frac{F({x,u_n } )}{\| {u_n } \|_E }\,dx}
 + \int_\Omega {h\frac{u_n }{\| {u_n } \|_E }\,dx} }\Bigg)\\
&\leq \limsup_{n \to + \infty }
 \frac{I(u_n)}{\| {u_n } \|_E }\\
&\leq \limsup_{n \to + \infty } \frac{\operatorname{const}}{\| {u_n } \|_E} = 0,
\end{align*}
which gives
\[
- \int_\Omega {f^{ + \infty } \phi _1 \,dx} + \int_\Omega {h\phi _1
\,dx}  \leq 0,
\]
which contradicts (H2). \smallskip

\noindent {\bf Case 2}: Assume that $v_n \to -\phi_1$.
Dividing \eqref{(8)} by $\|u_n\|_E $ we get
\begin{align*}
  \int_\Omega {f_{ - \infty } \phi _1 \,dx} - \int_\Omega {h\phi _1 \,dx}
&= \lim_{n \to + \infty }
 \Bigg({ - \int_\Omega {\frac{F({x,u_n } )}{\| {u_n } \|_E}\,dx}
 + \int_\Omega {h\frac{u_n}{\| {u_n } \|_E }\,dx} }\Bigg)\\
&\leq \limsup_{n \to + \infty }\frac{I(u_n)}{\| {u_n } \|_E}\\
&\leq \limsup_{n \to + \infty } \frac{\operatorname{const}}{\| {u_n } \|_E} = 0,
\end{align*}
which gives
\[
\int_\Omega {f_{ - \infty } \phi _1 \,dx} - \int_\Omega {h\phi _1
\,dx} \leq 0,
\]
which contradicts (H2).
\end{proof}

\begin{proof}[Proof of Theorem \ref{dlc1}]
The coerciveness and the Palais-Smale condition are enough to prove that
$I$ attains its proper infimum in Banach space $E$
(see Theorem \ref{bd:tmp}), so that \eqref{btbd} has at least a
weak solution in $E$. The proof is complete.
\end{proof}

\subsection*{Acknowledgments}
The authors wish to express their gratitude to the anonymous referee
for a number of valuable comments and suggestions which help us to
improve the presentation of the present paper from line to line.
This work is dedicated to the first author's father on the occasion
of his fiftieth birthday.

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\end{document}