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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 293, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/293\hfil Existence of solutions]
{Existence of solutions for quasilinear elliptic equations
involving a nonlocal term}

\author[M. F\u{a}rc\u{a}\c{s}eanu,  D. Stancu-Dumitru \hfil EJDE-2015/??\hfilneg]
{Maria F\u{a}rc\u{a}\c{s}eanu, Denisa Stancu-Dumitru}

\address{Maria F\u{a}rc\u{a}\c{s}eanu \newline
Department of Mathematics, University of Craiova,
200585 Craiova, Romania.\newline
Research group of the project PN-II-ID-PCE-2012-4-0021,
``Simion Stoilow" Institute of Mathematics of the Romanian Academy,
P.O. Box 1-764, 010702 Bucharest, Romania}
\email{farcaseanu.maria@yahoo.com}

\address{Denisa Stancu-Dumitru \newline
Research group of the project PN-II-ID-PCE-2011-3-0075,
``Simion Stoilow" Institute of Mathematics of the Romanian Academy,
P.O. Box 1-764, 010702 Bucharest, Romania}
\email{denisa.stancu@yahoo.com}

\thanks{Submitted April 15, 2015. Published November 30, 2015.}
\subjclass[2010]{35J60, 35J70, 35D30, 47J30, 58E05}
\keywords{Quasilinear elliptic equation; nonlocal term; weak solution;
\hfill\break\indent Schauder's fixed point theorem; critical point}

\begin{abstract}
 This article establishes the existence of solutions for a partial
 differential equation involving a quasilinear  elliptic operator
 and a nonlocal term. The proofs of the main results are based on
 Schauder's fixed  point theorem combined with  variational arguments.
\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}\label{sectia1}

Let $\Omega\subset \mathbb{R}^N$ denote a bounded  domain with
smooth boundary $\partial \Omega$,  and  $\nu$ denote the outward
unit normal to $\partial\Omega$. We consider the problem
\begin{equation}\label{pb1}
\begin{gathered}
-a\Big(\int_{\Omega} u(x)\,dx \Big)\operatorname{div}(a(u(x))\nabla u(x))+u(x)=0,
\quad  x\in \Omega,\\
a(u(x))\frac{\partial u}{\partial \nu}(x)=g(x),\quad
  x\in \partial\Omega,
\end{gathered}
\end{equation}
where $a:\mathbb{R}\to\mathbb{R}$  is a continuous function for
which there exist two constants $a_1,a_2\in(0,\infty)$ such that
\begin{equation}\label{CondA}
0<a_1\leq a(t)\leq a_2<\infty,\quad \forall t\in\mathbb{R}\,,
\end{equation}
and $g:\partial\Omega\to \mathbb{R}$ is a function satisfying
\begin{equation}\label{CondG}
g\in L^2(\partial \Omega).
\end{equation}
According to \cite[p. 160]{C2}, a physical motivation for studying
equations of type \eqref{pb1} comes from the fact that the
diffusion of the temperature in a material has a velocity given by
the \emph{Fourier law}
$$
\vec{v}=-a\nabla u\,,
$$
where $a$ is a constant proper to each material. Imagining a
material for which the constant is not the same for temperatures
between $0^o$ and $200^o$, i.e. it depends on the temperature of
the material itself, a more realistic Fourier law can be written
as
$$
\vec{v}=-a(u)\nabla u\,.
$$
This last relation can lead to equations of type \eqref{pb1} since
the expression above appears in the divergence operator from
\eqref{pb1}. For more physical motivations concerning problems of
type \eqref{pb1} the reader may also consult \cite[Chapter 1]{C1}.


On the other hand, note that the presence of function $a$
depending on $u$ in the divergence form from equation \eqref{pb1}
 represents the main difficulty in analysing the existence of
solutions for this problem since it does not enable one to
associate to the problem a so called \emph{energy functional} whose
critical points would offer weak solutions to our equation. For
that reason even if our treatment is in part variational we have
to combine it with a fixed point argument offered by Schauder's
fixed point theorem. Moreover, the presence of the nonlocal term
$a(\int_{\Omega} u(x)\,dx)$ allows us to be able to control the
number of solutions for our equation. Actually, as we will see in
the next section, in the particular case when
$\int_{\partial\Omega} g\,d\sigma(x)=1$ the number of fixed
points of function $a$ gives the number of solutions of equation
\eqref{pb1}, and, thus, we may have a unique solution of the
equation, a finite number of solutions or infinitely many
solutions. Moreover, if $\int_{\partial\Omega} g\,d\sigma(x)=1$
we can prescribe the number of solutions of our equation just by
prescribing the number of fixed points of $a$. Finally, note that
condition \eqref{CondA} on function $a$ was first introduced in
the pioneering paper by Arcoya \& Boccardo \cite{AB} but it was no
longer assumed in subsequent papers by Filippucci \cite{F1,F2}.
For other interesting results related to nonlocal problems we also 
refer to  \cite{MR1} and \cite{MR2}.

\section{Main result}

In this article we are interested in analyzing the existence and
multiplicity of weak solutions for problems of type \eqref{pb1}.
We start by recalling the definition of a weak solution for
problem \eqref{pb1}.

\begin{definition}\label{denisa162443} \rm
We say that $u\in H^1(\Omega)$ is a \emph{weak solution} of problem
\eqref{pb1} if
\begin{equation}\label{WS}
a\Big(\int_{\Omega} u(x) \,dx\Big)\Big[\int_{\Omega} a(u(x))
\nabla u\nabla \varphi\,dx-\int_{\partial\Omega} g\varphi\,d\sigma(x)\Big]
+\int_{\Omega} u \varphi\; dx=0, 
\end{equation}
for all $\varphi\in H^1(\Omega)$.
\end{definition}


The main result of this article read as follows.
 
\begin{theorem}\label{th1}
Assume conditions \eqref{CondA} and \eqref{CondG} are fulfilled.
If $ \int_{\partial\Omega} g(x)\,d\sigma(x)\neq 0$
then  \eqref{pb1} has as many weak solutions as equation
$ a(\mu) \int_{\partial\Omega} g\,d\sigma(x) =\mu$
while if   $ \int_{\partial\Omega} g(x)\,d\sigma(x)=0$ then 
 \eqref{pb1} has at least a weak solution.
\end{theorem}

To prove Theorem \ref{th1} we establish first the
following result whose proof will be given in the next section.

\begin{theorem}\label{th2}
Assume conditions \eqref{CondA} and \eqref{CondG} are fulfilled.
Then, for each $\mu\in \mathbb{R}$ fixed, the problem
\begin{equation}\label{pb3}
\begin{gathered}
-a(\mu)\operatorname{div}(a(u(x))\nabla u(x))+u(x)=0,\quad x\in \Omega,\\
a(u(x))\frac{\partial u}{\partial \nu}(x)=g(x),\quad  x\in
\partial\Omega
\end{gathered}
\end{equation}
has a weak solution $u_{\mu}\in H^1(\Omega)$, that is $u_{\mu}\in
H^1(\Omega)$ satisfies
\[
a(\mu)\Big[\int_{\Omega} a(u_{\mu}(x)) \nabla u_{\mu}\nabla \varphi\,dx
-\int_{\partial\Omega} g\varphi\,d\sigma(x)\Big]
+\int_{\Omega} u_{\mu} \varphi\,dx=0, 
\]
for all  $\varphi\in H^1(\Omega)$.
\end{theorem}

Now, we are ready to give the proof of Theorem \ref{th1}.

\begin{proof}[Proof of Theorem \ref{th1}]
 First, note that if $u\in H^1(\Omega)$ is a weak solution of  \eqref{pb1}, 
testing in \eqref{WS} with $\varphi=1$  we obtain
\begin{equation}\label{denEq}
a\Big(\int_{\Omega} u(x) \,dx\Big) 
\int_{\partial\Omega} g\,d\sigma(x)=\int_{\Omega} u(x) \,dx\,.
\end{equation}

On the one hand, if $  \int_{\partial\Omega} g\,d\sigma(x)\neq 0$, then by 
\eqref{denEq} it follows that
 $\mu:=\int_{\Omega} u(x) \,dx$ is a
solution of the equation
\begin{equation}\label{EqA}
a(\mu) \int_{\partial\Omega} g\,d\sigma(x) =\mu\,.
\end{equation}
Next, let  $\mu$ be a solution of the equation 
$a(\mu) \int_{\partial\Omega} g\,d\sigma(x) =\mu$. Then by
Theorem \ref{th2}  there exists $u_\mu\in H^1(\Omega)$ a weak
solution of the problem
\begin{gather*}
-a(\mu)\operatorname{div}(a(u(x))\nabla u(x))+u(x)=0,\quad  x\in \Omega,\\
a(u(x))\frac{\partial u}{\partial \nu}(x)=g(x),\quad  
x\in \partial\Omega
\end{gather*}
or
\begin{equation}\label{WS2}
a(\mu)\Big[\int_{\Omega} a(u_\mu(x)) \nabla u_\mu\nabla \varphi\,dx
-\int_{\partial\Omega} g\varphi\,d\sigma(x)\Big]
+\int_{\Omega} u_\mu \varphi\,dx=0, 
\end{equation}
for all $\varphi\in H^1(\Omega)$.
Testing in \eqref{WS2} with $\varphi=1$ we obtain
$$
a(\mu) \int_{\partial\Omega} g\,d\sigma(x) 
=\int_{\Omega} u_\mu\,dx\,.
$$ 
Since $\mu$ is a solution of the equation 
$ a(\mu) \int_{\partial\Omega} g\,d\sigma(x) =\mu$,
we find that
$\mu=\int_{\Omega} u_\mu\,dx$, and thus, 
$u_\mu$ is a solution of\eqref{pb1}.

On the other hand, if $\int_{\partial\Omega} g\,d\sigma(x)=0$ then 
relation \eqref{denEq} reads $\int_{\Omega} u(x)\,dx=0$. Thus, in this situation
we should seek  solutions of problem \eqref{pb1} in
$$
V:=\big\{v\in H^1(\Omega);\; \int_{\Omega} v(x)\,dx=0\big\}.
$$
Recall  that $V$ is a closed and convex subspace of $H^1(\Omega)$
and $H^1(\Omega)=V\oplus \mathbb{R}$. Thus, in this particular
case, $u\in V$ is a solution for problem \eqref{pb1} if $u$
satisfies
\begin{equation}\label{WS1}
a(0)\Big[\int_{\Omega} a(u(x)) \nabla u\nabla \varphi\,dx
-\int_{\partial\Omega} g\varphi\,d\sigma(x)\Big]
+\int_{\Omega} u \varphi\,dx=0, \quad \forall \varphi\in V.
\end{equation}
Therefore, if $  \int_{\partial\Omega} g\,d\sigma(x)=0$ the study of 
the existence of solutions for  \eqref{pb1} reduces to the study 
of the  existence of solutions for the problem
\begin{equation}\label{Eq0}
\begin{gathered}
-a(0)\operatorname{div}(a(u(x))\nabla u(x))+u(x)=0,\quad  x\in \Omega,\\
a(u(x))\frac{\partial u}{\partial \nu}(x)=g(x),\quad
  x\in \partial\Omega.
\end{gathered}
\end{equation}
This case can be treated in a similar manner as the proof of
Theorem \ref{th2} just by replacing $a(\mu)$ in \eqref{pb3},  by
$a(0)$ and analyzing the resulting problem in $V$ instead of
$H^1(\Omega)$. This completes the proof.
\end{proof}  

\begin{remark} \label{rmk1} \rm
In the particular case when 
$$  
\int_{\partial\Omega} g\,d\sigma(x)=1,
$$
by Theorem \ref{th1} we deduce that
we can prescribe the number of solutions of problem \eqref{pb1}
just by prescribing the number of fixed points of function $a$,
and thus,  we may have a unique solution of the problem
\eqref{pb1}, a finite number of solutions or infinitely many
solutions.

Next we point out a few examples of situations which can occur:

(1) If $a:[1,2]\to[1,2]$ with $a(t)=t$ for each $t\in[1,2]$
then problem \eqref{pb1} possesses infinitely many solutions
corresponding  to each point  $t\in[1,2]$ which are all fixed
points of function $a$.
\smallskip

(2) If $a:[1,2]\to[1,2]$ with $a(t)=3-t$ for each
$t\in[1,2]$ then problem \eqref{pb1} possesses a unique solution
corresponding to the unique fixed point of function $a$, namely
$t=3/2$.
\smallskip

(3) If $a:[1,2]\to[1,2]$ with
$$
a(t)=\begin{cases}
\frac{7}{3}-t ,& \text{if }  t\in [1,\frac{4}{3}],\\
3t-3, & \text{if }  t\in [\frac{4}{3},\frac{5}{3}],\\
\frac{16}{3}-2t, & \text{if } t\in [\frac{5}{3},2],
\end{cases}
$$
then problem \eqref{pb1} possesses exactly three solutions,
corresponding  to the  fixed points of function $a$ from the set
$\{\frac{7}{6}, \frac{3}{2}, \frac{16}{9}\}$.
\smallskip

(4) If $a:[1,2]\to[1,2]$ with
$$
a(t)=\begin{cases}
\frac{13}{6}-t, &\text{if } t\in[1,\frac{7}{6}],\\
 2t-\frac{4}{3}, & \text{if } t\in\left[\frac{7}{6},\frac{8}{6}\right],\\
t, & \text{if } t\in[\frac{8}{6},\frac{9}{6}],\\
6-3t, & \text{if } t\in[\frac{9}{6},\frac{10}{6}],\\
1, & \text{if } t\in[\frac{10}{6},\frac{11}{6}],\\
6t-10, &  \text{if } t\in[\frac{11}{6},2],
\end{cases}
$$ 
then  \eqref{pb1} possesses infinitely many
solutions, corresponding on the one hand  to the fixed points of
function $a$ from the interval $[\frac{8}{6},\frac{9}{6}]$ plus
two other solutions corresponding to the isolated fixed points of
function $a$ from the set $\{13/12,2\}$.
\end{remark}

\section{Proof of Theorem \ref{th2}}\label{Sd}

Fix $\mu\in\mathbb{R}$. The main tool in proving Theorem
\ref{th2} will be Schauder's fixed point theorem (see
\cite[Theorem 3.21]{ambrosetti}).


\begin{theorem}[Schauder's Fixed Point Theorem] \label{thm3.1}
 Assume that $K$ is a compact and convex subset of the Banach space $B$ and
$S:K\to K$ is a continuous map. Then $S$ possesses a fixed
point.
\end{theorem}

We will give the proof of Theorem \ref{th2} only in the case when 
$\int_{\partial \Omega} g\,d\sigma\neq 0$. The case 
$\int_{\partial \Omega} g\,d\sigma=0$ can be treated similarly
with the difference that we have to take $\mu=0$ this time and consider  
the weak formulation of the resulting problem on  
$\{u\in H^1(\Omega): \int_{\Omega} u(x)\,dx=0\}$ which is a closed 
subspace of $H^1(\Omega)$ (see the last part of the proof of Theorem \ref{th1} 
for more details).
We start by establishing some auxiliary results which will  be
useful in obtaining the conclusion of Theorem \ref{th2}.

\begin{lemma}\label{L1}
For each $v\in L^2(\Omega)$, the problem
\begin{equation}\label{pb4}
\begin{gathered}
-a(\mu)\operatorname{div}(a(v(x))\nabla u(x))+u(x)=0,\quad  x\in \Omega,\\
a(v(x))\; \frac{\partial u}{\partial \nu}(x)=g(x),\quad 
  x\in \partial\Omega
\end{gathered}
\end{equation}
has a weak solution $u\in H^1(\Omega)$, i.e. $u$ satisfies
\begin{equation}\label{WS3}
a(\mu)\Big[\int_{\Omega} a(v(x)) \nabla u(x)\nabla \varphi(x)\,dx
-\int_{\partial\Omega} g(x)\varphi(x)\,d\sigma(x)\Big]
+\int_{\Omega} u(x) \varphi(x)\,dx=0, 
 \end{equation}
for all  $\varphi\in H^1(\Omega)$.
\end{lemma}

\begin{proof}
 Fix $v\in L^2(\Omega)$. By hypotheses \eqref{CondA} we obtain
$a(v)\in L^{\infty}(\Omega)$.
Consider the energy functional associated with \eqref{pb4},
$J:H^1(\Omega)\to\mathbb{R}$ defined by
$$ 
J(u)=a(\mu)\int_{\Omega} \frac{a(v)}{2}  |\nabla u|^2\,dx
+\frac{1}{2}\int_{\Omega} u^2\,dx-a(\mu)
\int_{\partial\Omega} g u\,d\sigma(x).
$$
Standard arguments imply that $J\in C^1(H^1(\Omega),\mathbb{R})$
with the derivative given by
\[
\langle J'(u),\varphi\rangle=a(\mu)\int_{\Omega} a(v)  \nabla u
 \nabla \varphi\,dx
+ \int_{\Omega} u \varphi\,dx-a(\mu)\int_{\partial\Omega} g
\varphi\,d\sigma(x),
\]
for all  $u,\varphi\in H^1(\Omega)$.
Thus, the weak solutions of \eqref{pb4} are exactly the
critical points of $J$.

For each $u\in H^1(\Omega)$, using the fact that $H^1(\Omega)$ is
continuously embedded in $L^2(\partial\Omega)$ (see, e.g.
\cite[Theorem 5.6.1]{ABM}) and conditions \eqref{CondA} and
\eqref{CondG} holds, we deduce that
\begin{align*}
J(u)
&\geq \frac{a_1^2}{2}\int_{\Omega} |\nabla u|^2\,dx
 +\frac{1}{2} \int_{\Omega} u^2\,dx
 - a_2 \|g\|_{L^2(\partial\Omega)} \|u\|_{L^2(\partial\Omega)}\\
&\geq \min\{\frac{a_1^2}{2},\frac{1}{2}\}
 \|u\|^2_{H^1(\Omega)}-a_2 C \|g\|_{L^2(\partial\Omega)}  \|u\|_{H^1(\Omega)}\,,
\end{align*}
where $C$ is a positive constant. The above estimates show that
$J$ is coercive. On the other hand, it is standard to check that
$J$ is weakly lower semi-continuous. Then, the Direct Method in
the Calculus of Variations (see, e.g. \cite[Theorem 1.2]{S})
guarantees the existence of a global minimum point of $J$, 
$u\in H^1(\Omega)$ and consequently a weak solution of 
\eqref{pb4}. The proof of Lemma \ref{L1} is complete. 
\end{proof}

Next, for each $v\in L^2(\Omega)$  let $u=T(v)\in H^1(\Omega)$ be
the weak solution of  \eqref{pb4} given by Lemma \ref{L1}.
Thus, we can actually introduce a mapping
$$
T:L^2(\Omega)\to H^1(\Omega)
$$
associating to each $v\in L^2(\Omega)$ the weak solution of
problem \eqref{pb4}, $T(v)\in H^1(\Omega)$.

\begin{lemma}\label{L2}
There exists a universal constant $\mathcal{C}>0$, which does not
depend on $\mu$ or $v$, such that
\begin{equation}\label{Ineq1}
\int_{\Omega} |\nabla T(v)|^2\,dx+\int_{\Omega} |T(v)|^2\,dx\leq
\mathcal{C},\quad \forall  v\in L^2(\Omega).
\end{equation}
\end{lemma}

\begin{proof}
 Since $T(v)$ is a weak solution of  \eqref{pb4},
taking $\varphi=T(v)$ in \eqref{WS3} we find that
$$
a(\mu)\int_{\Omega} a(v(x)) |\nabla T(v)|^2\,dx
+\int_{\Omega} |T(v)|^2\,dx
=a(\mu)\int_{\partial\Omega} g T(v)\,d\sigma(x).
$$ 
Using relation \eqref{CondA}, H\"older's
inequality and the fact that $H^1(\Omega)$ is continuously
embedded in $L^2(\partial\Omega)$  we deduce
\begin{align*}
\min\left\{\frac{a_1^2}{2},\frac{1}{2}\right\} 
\|T(v)\|^2_{H^1(\Omega)}
&= \min\big\{\frac{a_1^2}{2},\frac{1}{2}\big\} 
\Big(\int_{\Omega} |\nabla T(v)|^2\,dx+\int_{\Omega} |T(v)|^2\,dx\Big)\\
&\leq a_2 \|g\|_{L^2(\partial\Omega)} \|T(v)\|_{L^2(\partial\Omega)}\\
&\leq a_2  D  \|g\|_{L^2(\partial\Omega)}
\|T(v)\|_{H^1(\Omega)},
\end{align*}
where $D$ is a positive constant. Leting
\[
\mathcal{C}:=\Big(\frac{ D a_2\|g\|_{L^2(\partial\Omega)}}
{\min\{\frac{a_1^2}{2},\frac{1}{2}\}}\Big)^2
\]
 we obtain inequality \eqref{Ineq1}. The proof  is complete.
\end{proof}

\begin{lemma}\label{L3}
The mapping $T:L^2(\Omega)\to H^1(\Omega)$ is continuous.
\end{lemma}

\begin{proof} 
Let  $\{v_n\}\subset L^2(\Omega)$ and $v\in L^2(\Omega)$
such that $\{v_n\}$ converges strongly to $v$ in $L^2(\Omega)$.
Set $u_n:=T(v_n)$ for any positive integer $n$.

By Lemma \ref{L2} we infer that
$$ 
\int_{\Omega} \left(|\nabla u_n|^2+u^2_n\right)\,dx
=\int_{\Omega} \left(|\nabla T(v_n)|^2+
|T(v_n)|^2\right)\,dx
\leq \mathcal{C},\quad \forall  n;
$$
that is, the sequence $\{u_n\}$ is bounded in $H^1(\Omega)$. 
It follows that there exists $u\in H^1(\Omega)$ such that, 
up to a subsequence still denoted by $\{u_n\}$,  converges weakly to $u$ in
$H^1(\Omega)$  and by Rellich-Kondrachov theorem (see, e.g.
\cite[Theorem 5.5.2]{ABM}) we deduce that $\{u_n\}$ converges
strongly to $u$ in $L^2(\Omega)$.  On the other hand, we have
$u_n$ is a weak solution of problem \eqref{pb4} and thus by
\eqref{WS3} we obtain
\begin{equation}\label{rCdenisa}
a(\mu) \int_{\Omega} a(v_n) \nabla u_n \nabla \varphi\,dx
+\int_{\Omega} u_n \varphi\,dx=a(\mu)\int_{\partial\Omega}
g \varphi\,d\sigma(x),
\end{equation}
for all  $\varphi\in H^1(\Omega)$ and all $n$.

Since $\{v_n\}$ converges strongly to $v$ in $L^2(\Omega)$, it
follows that $v_n(x)\to v(x)$ a.e. $x\in \Omega$, too.
Combining that fact with the one that function $a$ is continuous
a.e. on $\mathbb{R}$, we find
\begin{equation}\label{Cv}
a(v_n(x))\to a(v(x))\quad\text{for  a.e. } x\in \Omega.
\end{equation}
Moreover, since $\{u_n\}$ converges weakly to $u$ in $H^1(\Omega)$
we deduce that
\begin{equation}\label{Wconv}
\{\nabla u_n\}\text{ converges  weakly to $\nabla u$  in }
  (L^2(\Omega))^N.
\end{equation}
Lebesgue's dominated convergence theorem (see, e.g. \cite[Theorem
4.2]{B}) and \eqref{Cv} imply that
 \begin{equation}
\{a(v_n)\nabla \varphi\}\text{ converges  strongly to
$a(v)\nabla \varphi$  in }  (L^2(\Omega))^N, \;
\forall   \varphi\in H^1(\Omega).
\end{equation}
Thus, we deduce that
$$
\int_{\Omega} a(v_n) \nabla u_n\nabla \varphi\,dx\to
 \int_{\Omega} a(v) \nabla u\nabla \varphi\,dx,\quad \forall  \varphi\in
H^1(\Omega).
$$ 
In particular, for $\varphi=u$ we have
\begin{equation}\label{Conv}
\int_{\Omega} a(v_n) \nabla u_n \nabla u\,dx\to
\int_{\Omega} a(v) |\nabla u|^2\,dx.
\end{equation}
Taking $\varphi=u_n-u$ in \eqref{rCdenisa} and taking 
into account the above pieces of information we also find that
\begin{equation}
\int_{\Omega} a(v_n) \nabla u_n (\nabla u_n-\nabla u)\,dx=o(1)
\end{equation}
and consequently,
\begin{equation}\label{Eq1}
\int_{\Omega} [a(v_n)-a(v)] \nabla u_n (\nabla u_n-\nabla u)\,dx
+\int_{\Omega} a(v) \nabla u_n (\nabla u_n-\nabla u)\,dx=o(1).
\end{equation}
By \eqref{Cv} and the fact that $\{|\nabla u_n|^2\}$ is a bounded
sequence in $L^1(\Omega)$ we obtain by H\"older's inequality that
\begin{equation}\label{Eq2}
\big|\int_{\Omega} [a(v_n)-a(v)] |\nabla u_n|^2\,dx\big|
\leq \|a(v_n)-a(v)\|_{L^{\infty}(\Omega)} \|\nabla
u_n\|^2_{L^2(\Omega)} \to 0
\end{equation}
 as $ n\to\infty$.
Then \eqref{Wconv}, \eqref{Conv} and \eqref{Eq2} yield
\begin{equation}\label{Eq3}
\int_{\Omega} [a(v_n)-a(v)]  \nabla u_n (\nabla u_n-\nabla u)\,dx=o(1).
\end{equation}
By \eqref{Eq1} and \eqref{Eq3} we find
\begin{equation}\label{Eq4}
\int_{\Omega} a(v)\;  \nabla u_n (\nabla u_n-\nabla u)\,dx=o(1).
\end{equation}
We deduce that
\begin{equation}\label{Equality}
\begin{aligned}
&\int_{\Omega} a(v) |\nabla u_n-\nabla u|^2\,dx\\
&=\int_{\Omega} a(v) |\nabla u_n|^2\,dx
-2\int_{\Omega} a(v) \nabla u_n\nabla u\,dx
+\int_{\Omega} a(v) |\nabla u|^2\,dx.
\end{aligned}
\end{equation}
By \eqref{Eq4} we infer that
\[
\lim_{n\to \infty} \int_{\Omega} a(v) |\nabla u_n|^2\,dx
= \lim_{n\to \infty} \int_{\Omega}a(v) \nabla u_n\nabla u\,dx
=\int_{\Omega} a(v) |\nabla u|^2\,dx
\]
and using  \eqref{Equality} we finally obtain
 $$
\int_{\Omega} a(v) |\nabla (u_n-u)|^2\,dx=o(1)
$$
 which implies that 
$$
\int_{\Omega} |\nabla (u_n-u)|^2\,dx=o(1).
$$
 Moreover, taking into account that $\{u_n\}$ converges strongly to $u$
in $L^2(\Omega)$ we conclude that $\{u_n\}$ converges
 strongly to $u$ in $H^1(\Omega)$, that means  application $T$
 is continuous.  The proof  is complete.
\end{proof}

\begin{remark} \label{R4} \rm
Since $H^1(\Omega)$ is compactly embedded in $L^2(\Omega)$, that is
the inclusion operator $i:H^1(\Omega)\to L^2(\Omega)$
is compact, it follows by Lemma \ref{L3} that the operator
$S:L^2(\Omega)\to L^2(\Omega)$ defined by $S=i\circ T$ is
compact.
\end{remark}



\begin{proof}[Proof of Theorem \ref{th2}]
  Let $\mathcal{C}$ be the positive constant given by
Lemma \ref{L2}. We have
$$
\int_{\Omega} |\nabla S(v)|^2\,dx+\int_{\Omega} |S(v)|^2\,dx
\leq \mathcal{C},\;\; \forall \; v\in L^2(\Omega).
$$
In particular, 
$$
\int_{\Omega} |S(v)|^2\,dx\leq \mathcal{C},\quad 
\forall  v\in L^2(\Omega).
$$ 
In $L^2(\Omega)$, define the set
$$
B_{\mathcal{C}}(0):=\big\{v\in L^2(\Omega):
 \int_{\Omega} |v(x)|^2\,dx\leq \mathcal{C}\big\}.
$$
Clearly,  $B_{\mathcal{C}}(0)$ is a convex, closed subset of
$L^2(\Omega)$ and $S(B_{\mathcal{C}}(0))\subset
B_{\mathcal{C}}(0)$. By Remark \ref{R4}  it follows that
$S(B_{\mathcal{C}}(0))$ is relatively compact in $L^2(\Omega)$.

Finally, by Lemma \ref{L3} and Remark \ref{R4}, we  deduce that
$S:S(B_{\mathcal{C}}(0))\to S(B_{\mathcal{C}}(0))$ is a
continuous map. Hence, we can apply the Schauder's fixed point
theorem (Theorem \ref{thm3.1})
to obtain that $S$ possesses a fixed point. This gives us
a weak solution of problem \eqref{pb3} and thus the proof of
Theorem \ref{th2} is finally complete.
\end{proof}



\subsection*{Acknowledgments}
 M. F\u{a}rc\u{a}\c{s}eanu was partially supported by  CNCS-UEFISCDI Grant
No. PN-II-ID-PCE-2012-4-0021 ``Variable Exponent Analysis: Partial
Differential Equations and Calculus of Variations". 
D. Stancu-Dumitru was  partially
supported by  grant CNCSIS-UEFISCSU PN-II-ID-PCE-2011-3-0075
``Analysis, Control and Numerical Approximations of Partial
Differential Equations".


\begin{thebibliography}{99}


\bibitem{ambrosetti} Ambrosetti, A.; Malchiodi, A.;
 \emph{Nonlinear Analysis and Semilinear Elliptic Problems},
Cambridge University Press, Cambridge, 2007.

\bibitem{AB} Arcoya, D.; Boccardo, L.;
Critical points for multiple integrals of the calculus of variations,
\emph{Arch. Rational Mech. Anal.} \textbf{134} (1996), 249--274.

\bibitem{ABM}  Attouch, H.;  Buttazzo, G.;  Michaille, G.;
\emph{Varitional Analysis in Sobolev and BV Spaces. Applications to PDEs and
Optimization}, MPS-SIAM Series on Optimization, 2006.

\bibitem{B}  Brezis, H.;
 \emph{Functional Analysis, Sobolev Spaces and Partial Differential Equations}, 
Springer, 2011.

\bibitem{C1} Chipot, M.;
 \emph{Elements of Nonlinear Analysis}, Birh\"auser, 2000.

\bibitem{C2} Chipot, M.; \emph{Elliptic Equations: An Introductory Course}, 
Birh\"auser, 2009.

\bibitem{F1} Filippucci, R.;
 Existence of global solutions of elliptic systems, 
\emph{J. Math. Anal. Appl.} \textbf{293} (2004), 677--692.

\bibitem{F2} Filippucci, R.;
 Entire radial solutions of elliptic systems and inequalities of the mean 
curvature type, \emph{J. Math. Anal. Appl.} \textbf{334} (2007), 604--620.

\bibitem{MR1} Molica Bisci, G.; R\u{a}dulescu, V.;
 Applications of Local Linking to Nonlocal Neumann Problems, 
\emph{Commun. Contemp. Math.} \textbf{17} (1) (2015), 17 pages, ID 1450001.

\bibitem{MR2} Molica Bisci, G.;  Repov\v{s}, D.;
 On doubly nonlocal fractional elliptic equations, \emph{Rend. Lincei Mat. Appl.} 
\textbf{26} (2015), 161--176.

\bibitem{S} Struwe, M.;
\emph{Variational Methods: Applications to
Nonlinear Partial Differential Equations and Hamiltonian Systems},
Springer, Heidelberg, 1996.

\end{thebibliography}

\end{document}