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\address{Mostafa Allaoui \newline
University Mohamed I, Faculty of sciences, Department of
Mathematics,\newline Oujda, Morocco} \email{ allaoui19@hotmail.com}

\address{Abdelrachid El Amrouss \newline
University Mohamed I, Faculty of sciences, Department of
Mathematics,\newline Oujda, Morocco} \email{elamrouss@fso.ump.ma,
elamrouss@hotmail.com}

\address{Anass Ourraoui \newline
University Mohamed I, Faculty of sciences, Department of
Mathematics,\newline Oujda, Morocco} \email{anas.our@hotmail.com}
\keywords{ p(x)-laplacian, Weak solutions, variational
 methods}
 \begin{document}
\title[Existence of solutions]{ Existence and uniqueness of solution for
 $p(x)$-Laplacian problems}
\author{Mostafa Allaoui, Abdelrachid El Amrouss, Anass Ourraoui}  % in alphabetical order
\begin{abstract}
 This paper shows the existence of weak solution of
a problem which involves the $p(x)$-laplacian with some different boundary conditions. The proof
of the result is made by Browder Theorem.

\end{abstract}
\maketitle
%%%%%%%%%%%%%%%%%%%%%%=========================================

\section{Introduction}

At the turn of the millennium, a large number of papers is scattered
to study of elliptic equations and variational problems with
variable exponent, it is of considerable importance in the theory of
partial differential equations. Some of these problems come from
different areas of applied mathematics and physics such as Micro
Electro-Mechanical systems, surface diffusion on solids or image
processing and restoration... For more inquiries on modeling
physical phenomena involving $p(x)$-growth condition we refer to
\cite{2,3,4,7}.


In this work, we consider the following problems involving $p(x)$-laplacian


$\centerdot$ Dirichlet problem

\begin{equation}
\begin{gathered} -\Delta _{p(x)}u=f(x,u)\quad
\text{in }\Omega , \\
u=0\quad \text{on }\partial \Omega,
\end{gathered}  \label{e0}
\end{equation}


$\centerdot$ Neumann problem

\begin{equation}
\begin{gathered}- \Delta _{p(x)}u+a(x)|u|^{p(x)-2}u=f(x,u)\quad
\text{in }\Omega , \\
\frac{\partial u}{\partial \nu}=0\quad \text{on }\partial \Omega,
\end{gathered}  \label{e1}
\end{equation}


$\centerdot$ No flux problem

\begin{equation}
\begin{gathered} -\Delta _{p(x)}u=f(x,u)\quad
\text{in }\Omega , \\
u=constant\quad \text{on }\partial \Omega,\\
\int_{\partial\Omega}|\nabla u|^{p(x)-2}\frac{\partial u}{\partial
\nu}=0.
\end{gathered}  \label{e2}
\end{equation}


$\centerdot$ Steklov problem

\begin{equation}
\begin{gathered} \Delta _{p(x)}u=|u|^{p(x)-2}u\quad
\text{in }\Omega , \\
|\nabla u|^{p(x)-2}\frac{\partial u}{\partial \nu}=g(x,u)\quad
\text{on }\partial \Omega,
\end{gathered}  \label{e3}
\end{equation}


$\centerdot$ Robin problem

\begin{equation}
\begin{gathered} \Delta _{p(x)}u=f(x,u)\quad
\text{in }\Omega , \\
|\nabla u|^{p(x)-2}\frac{\partial u}{\partial
\nu}+\beta|u|^{p(x)-2}u=0\quad \text{on }\partial \Omega,
\end{gathered}  \label{e4}
\end{equation}
where $\Omega$ is a bounded domain in
$\mathbb{R}^N,\Delta_{p(x)}u=div(|\nabla u|^{p(x)-2}u)$ is the
$p(x)-$laplacian operator, $p\in C(\Omega),$ with
$1<p^-=\inf_{x\in\Omega}\leq p(x)\leq
\sup_{x\in\Omega}p(x)=p^+<\infty$ and $\frac{\partial u}{\partial
\nu}$ denotes the out normal derivative of u, $\lambda \in
\mathbb{R}$ and $\beta:\partial\Omega\rightarrow \mathbb{R}~\in
L^{\infty}(\partial\Omega)$ is real function with
$\beta^-=\inf_{x\in\partial\Omega}\beta(x)>0.$

%\includegraphics{kir.jpg}


Assume that

$(f_1)$ f is a carathéodory function which is decreasing with
respect to the second variable.

$(f_2)$ There exists  $b>0$ and $c>0$ such that $$|f(x,t|\leq
b(1+|t|^{q(x)})$$ and $$|g(x,t)\leq c(1+|t|^{r(x)})$$ where
$$\sup_{\Omega}q(x)=q^{+}< p^-,~~ for~~ all x\in\Omega$$ and
$$\sup_{\Omega}r(x)=r^{+}< p^-~~for ~~all ~~x\in\partial\Omega.$$

$(f_3)$  $f(x,0)\neq0,~~g(x,0)\neq0.$

We report our main result,
\begin{thm}
Suppose that f satisfy the conditions $(f_1),(f_2)$ and $(f_3).$ Then the
problems $(\ref{e0}-$ $\ref{e4})$ have a unique weak nontrivial
solution.
\end{thm}

When $p = 2$, the problem $(\mathcal{P})$ is a normal Schrodinger
equation which has been extensively studied. There are several
studies of the existence of solutions of $(\mathcal{P})$ on a
bounded domain of $\mathbb{R}^N.$ We mention the results obtained in
\cite{1},\cite{6} and \cite{9} for the case when p is constant. In recent
years, more and more attention is paid to the quasilinear elliptic
with a variable exponent. The main difficulty in the study of
p(x)-Laplacian equations arises from its inhomogeneity.

Define the operators I, J , L and K $:X \rightarrow X^*$ by

$$ \langle I(u),v\rangle=\int_{\Omega}|\nabla u|^{p(x)-2}\nabla u.\nabla vdx, $$
$$ \langle J(u),v\rangle= \int_{\Omega}|u|^{p(x)-2}uvdx, $$
$$\langle L(u),v\rangle\int_{\Omega}f(x,u)vdx,  $$
$$\langle K(u),v\rangle=\int_{\partial\Omega}g(x,u)vdx.$$
\begin{defn}

 let $X=W^{1,p(x)}_0(\Omega),W^{1,p(x)}(\Omega)$ or
$W^{1,p(x)}_0(\Omega)\oplus  \mathbb{R},u\in X.$

(i) We say that u is a weak solution of (\ref{e0}), (\ref{e1}) and
(\ref{e2}) if $$ \int_{\Omega}| \nabla u|^{p(x)-2}\nabla u.\nabla
v=\int_{\Omega}f(x,u)vdx$$ for all $v\in X.$

(ii) Let $u\in W^{1,p(x)}(\Omega),$ a weak solution u for Steklov
problem (\ref{e3}) provided
$$\int_{\Omega}| \nabla u|^{p(x)-2}\nabla u.\nabla
v+|u|^{p(x)-2}uv=\int_{\partial\Omega}g(x,u)v~dx$$ for all $v\in
W^{1,p(x)}(\Omega).$

(iii) Let $u\in W^{1,p(x)}(\Omega),$ u is a weak solution of the
Robin problem (\ref{e4}) if
$$ \int_{\Omega}| \nabla u|^{p(x)-2}\nabla u.\nabla
v+\int_{\partial\Omega}\beta|
u|^{p(x)-2}uvdx=\int_{\Omega}f(x,u)v~dx$$ for all $v\in
W^{1,p(x)}(\Omega).$
\end{defn}

\begin{thm}(cf.\cite{7})
Let A be a reflexive real Banach space. Moreover, let $A : X
\rightarrow X^* $ be an operator which is: bounded, demicontinuous,
coercive, and monotone on the space X . Then, the equation $A(u) =
f$ has at least one solution $u \in X$ for each $f \in X^*.$ If
moreover, A is strictly monotone operator, then the equation
$(\mathcal{P})$ has precisely one solution $u \in X$ for every $f\in
X^*.$
\end{thm}

Define the operator $A:~X\rightarrow X^*$ by

$T:= I+J-L,$ or $T=I+J-K.$

\begin{defn}
Let X be a real Hilbert space. An operator $I : X \rightarrow X^*$
verifies
\begin{equation}\label{ed}
\langle I(u)-I(v),u-v\rangle\geq0
\end{equation}

 for any $u,v\in X$ is called a
monotone operator. An operator I is called strictly monotone if for
$u\neq v$ the strict inequality holds in \ref{ed}. An operator I is
called strongly monotone if there exists $C > 0$ such that
$$\langle I(u)-I(v),u-v\rangle\geq C\parallel u-v\parallel^2 $$
for any $u,v\in X.$
\end{defn}

\section{Preliminary Notes }

In order to deal with the problem $(\mathcal{P})$, we need some
theory of variable exponent Sobolev Space. For convenience, we only
recall some basic facts which will be used later, we refer to
\cite{5} and references therein for more details.

Define  the variable exponent Lebesgue space $L^{p(x)}(\Omega)$,\\
$L^{p(x)}(\Omega)=\{u:\Omega\rightarrow \Omega~
measurable~:~\int_{\Omega}\mid u\mid^{p(x)}dx < \infty\}$ then
$L^{p(x)}(\Omega)$ endowed with the norm $$\mid
u\mid_{p(x)}=\inf\{\lambda>0~:~ \int_{\Omega}\mid
\frac{u}{\lambda}\mid^{p(x)}dx \leq1\}$$  becomes a Banach space
separable and reflexive space.

Define the variable exponent Sobolev space $W^{1,p(x)}(\Omega)$ by
$$W^{1,p(x)}(\Omega)=\{u\in L^{p(x)}(\Omega):\nabla u\in
L^{p(x)}(\Omega)\}$$ equipped with the norm
$$\parallel u\parallel=|u|_{p(x)}+| \nabla u|_{p(x)},$$
which is a separable reflexive Banach space.

\begin{pro}\label{prop1}
Set, $\rho(u)=\int_{\Omega}\mid \nabla u\mid^{p(x)} dx,$ if $u\in
W^{1,p(x)}(\Omega)$

we have

(1) $\parallel u\parallel\geq 1 \Rightarrow \parallel
u\parallel^{p^-}\leq \rho(u)\leq \parallel u\parallel^{p^+}.$

(2) $\parallel u\parallel\leq 1 \Rightarrow \parallel
u\parallel^{p^+}\leq \rho(u)\leq \parallel u\parallel{p^-}.$

(3) $\displaystyle\lim_{n\rightarrow\infty}\parallel
u_n\parallel=0(resp\rm~+\infty)~\Leftrightarrow
\displaystyle\lim_{n\rightarrow+\infty}\rho(u_n)=0~(resp\rm~+\infty).$
\end{pro}

\begin{rem}\label{rem1}
We have similar results (1) and (2) for $\rho_1(u)=\int_{\Omega}\mid
u\mid^{p(x)}dx$ as in the above.
\end{rem}
\begin{pro}\label{H}
For any $u\in L^{p(x)}(\Omega),~v\in L^{p'(x)}(\Omega),$ we have
$$ \mid\int_{\Omega} uv~dx\mid\leq 2 \mid u\mid_{p(x)}\mid v\mid_{p'(x)},$$
with $$\frac{1}{p(x)}+\frac{1}{p'(x)}=1.$$
\end{pro}

\begin{lem} \label{lem2.4}
 Assume that the boundary of $\Omega$ possesses the cone property and
$p\in C(\overline{\Omega})$ and $1\leq q(x)<p^{\ast}(x)$
 for $x \in \overline{\Omega}$, then there is a compact embedding
 $ W^{1,p(x)}(\Omega)\hookrightarrow L^{q(x)}(\Omega)$,
 where
  $$
  p^{\ast}(x)=  \begin{cases}
                 \frac{Np(x)}{N-p(x)}, & \text{if } p(x) < N; \\
                 +\infty , & \text{if } p(x)\geq N.
               \end{cases}
 $$
\end{lem}

 \begin{lem} \label{lem2.5}
 If $f:\overline{\Omega}\times\mathbb{R}\to\mathbb{R}$ is a carath\'eodory function
and
 $$
 | f(x,s)|\leq a(x)+b| s|^{\frac{p_1(x)}{p_{2}(x)}} , \quad
\forall (x,s)\in \overline{\Omega}\times\mathbb{R} ,
 $$
 where $p_1(x) ,p_{2}(x)\in C(\overline{\Omega}) $,
$ a(x) \in L^{p_{2}(x)}(\Omega)$, $p_{2}(x)>1$, $ p_{2}(x)>1$,
$a(x)\geq 0$ and
 $b\geq 0$ is a constant, then the Nemytskii operator from $L^{p_1(x)}(\Omega)$
to $L^{p_{2}(x)}(\Omega)$ defined by $N_{f}(u)(x)=f(x,u(x))$ is a
continuous and bounded operator.
 \end{lem}

  Define
\begin{gather*}
 p^{\partial}(x)=(p(x))^{\partial}:= \begin{cases}
 \frac{(N-1)p(x)}{N-p(x)}, & \text{if } p(x)< N, \\
 \infty, & \text{if } p(x)\geq N.
\end{cases}
\end{gather*}

 \begin{lem} \label{2.7}
 Assume that the boundary of $\Omega$ possesses the cone property and
$p\in C(\overline{\Omega})$ with $p^{-}>1,$ there is a compact
embedding $W^{1,p(x)}(\Omega)\hookrightarrow
L^{q(x)}(\partial\Omega)$, where $1\leq q(x)<p^{\partial}(x) $,
$\forall x\in \partial\Omega$.
 \end{lem}

\section{proof of the main result}

(A) I, J, K and L are bounded, in fact, let $\parallel
u\parallel_0\leq M,$

Since $I$ and J are the Fréchet derivative of the functional
$\int_{\Omega}\frac{1}{p(x)}(\mid \nabla u\mid^{p(x)}+\mid
u\mid^{p(x)})dx$ and $\int_{\Omega}\frac{1}{p(x)}\mid
u\mid^{p(x)}dx$ respectively, therefore $I$ and J are bounded.

We have the same deduction for
$\int_{\partial\Omega}\beta(x)|u|^{p(x)-2}u.$


Moreover, from proposition \ref{H} and lemma \ref{lem2.4}, there
exists $C_1>0$ such that

\begin{align*}
\parallel L(u)\parallel_{X^*}&=\sup_{\parallel v\parallel=1}\mid \langle K(u),v
\rangle\mid\\
&\leq \sup_{\parallel v\parallel=1}2\mid f\mid_{p'(x)} \mid v\mid_{p(x)}dx\\
&\leq C_1\mid f\mid_{p'(x)}.
\end{align*}

Similarly, in view of lemma \ref{2.7} there exists $C_2>0,$ such
that
 $$\parallel
K(u)\parallel_{X^*}\leq C_2\mid g\mid_{l^{p(x)}(\partial\Omega)}.$$

so K is a bounded operator.

(B)I ,J , K and L are continuous,

We have I and J are continuous operators because that are the
Fr\^{e}chet derivative of the functional
$\int_{\Omega}\frac{1}{p(x)}(\mid \nabla u\mid^{p(x)}+\mid
u\mid^{p(x)})dx$ and $\int_{\Omega}\frac{1}{p(x)}\mid
u\mid^{p(x)}dx$ respectively, and then $I$ J are continuous.


On the other hand,

 Let $(u_n)_n\subset X$ be a sequence such that $u_n\rightharpoonup
 u.$ Since there is a compact embedding of $X$ into
$L^{q(x)}(\Omega),$ there is a subsequence, noted also $(u_n)_n$,
such that $u_n\rightarrow u$ in $L^{q(x)}(\Omega).$ According to the
Krasnoselki's theorem, the Nemytskii operator

$$~~\begin{array}{rl}
& \mbox{$ N_f:~L^{q(x)}(\Omega) ~\rightarrow L^{\frac{q(x)}{q(x)-1}}(\Omega)$} \\
& \mbox{~~~$~~~~~~~~~~~~u~~~~~~~\mapsto ~~f(., ~u)$}\end{array}$$ is
continuous. Hence, $N_g(u_n)\rightarrow N_g(u)$ in
$L^{\frac{q(x)}{q(x)-1}}(\Omega).$ Using Holder's inequality and the
continuous embedding of $X$ into $L^{q(x)}(\Omega),$ we obtain
\begin{eqnarray}
\mid \langle  L(u_n)-L(u), v \rangle  \mid&=&\mid \int_{\Omega}
(f(x,u_n)-f(x,u))v(x)dx\mid\nonumber\\
&\leq& 2\parallel  N_f(u_n)-N_f(u)
\parallel_{\frac{q(x)}{q(x)-1}} \mid v(x)\mid_{q(x)}\nonumber \\
&\leq & C\mid N_f(u_n)-N_f(u)\mid_{\frac{q(x)}{q(x)-1}}
\parallel v\parallel.\nonumber
 \end{eqnarray}
Thus, $L(u_n)\rightarrow L(u).$

Further, it is known that the Nemytskii operator $N_g:~u\mapsto
g(x,u) $ is a continuous bounded operator from
$L^{r(x)}(\partial\Omega)$ into
$L^{\frac{r(x)}{r(x)-1}}(\partial\Omega),$ and analogously, K is
completely continuous.

(C) T is strongly monotone,

We set $$U_p=\{ x\in \Omega:p(x)\geq 2 \},$$
$$V_p=\{x\in \Omega: 1<p(x)<2 \}.$$
By the elementary inequalities,(cf.\cite{8}) we have $\forall x,y\in
\mathbb{R}^N$
$$\mid x-y\mid^\gamma\leq 2^\gamma(\mid x\mid^{\gamma-2}x-\mid
y\mid^{\gamma-2}y ).(x-y)~~ if~~\gamma\geq2,$$
$$\mid x-y\mid^2\leq\frac{1}{\gamma-1}(\mid x\mid+\mid y\mid)^{2-\gamma}
(\mid x\mid^{\gamma-2}x-\mid
y\mid^{\gamma-2}y).(x-y)~~if~~1<\gamma<2,$$ where $x.y$ denotes the
usual inner product in $\Omega.$ It follows that

\begin{eqnarray}
\langle (I+J)(u)-(I+J)(v),u-v \rangle&=&\int_{\Omega}[\{\mid \nabla
u\mid^{p(x)-2}\nabla u-\mid \nabla v\mid^{p(x)-2}\nabla
v)\}\nabla(u-v)\nonumber\\
&+&\{(\mid u\mid^{p(x)-2}u-\mid
v\mid^{p(x)-2}v)\}]dx\nonumber\\
&\geq& \frac{1}{2^{p^+}}\int_{U_p}[~\mid \nabla(u-v)\mid^{p(x)}+\mid
u-v\mid^{p(x)} ~]dx+\nonumber
\\ &&(p^--1)\int_{V_p}[~\mid \nabla(u-v)\mid^{p(x)}+\mid
u-v\mid^{p(x)} ~]dx.\nonumber
\end{eqnarray}
 From proposition 2.1, taking $c_0=\min \{\frac{1}{2p^+},{p^-}-1\}$, then we
have \begin{equation}\label{eq.8} \langle (I+J)(u)-(I+J)(v),u-v
\rangle\geq c_0\left\{\begin{array}{rl}
& \mbox{$\parallel u-v\parallel^{p^+}~~if~~0\leq\parallel u-v\parallel\leq1 ,$} \\
& \mbox{$ \parallel u-v\parallel^{p^-}~~if~~\parallel
u-v\parallel>1$}\end{array}\right.
\end{equation}
 hence $I+J$ is strongly monotone(cf.\cite{10}).

Since f is decreasing with respect to the second variable, then

$\langle
L(u)-L(v),u-v\rangle=\int_{\Omega}(f(x,u)-f(x,v))(u-v)dx\geq0.$

Also,

$\langle
K(u)-K(v),u-v\rangle=\int_{\partial\Omega}(g(x,u)-g(x,v))(u-v)dx\geq0.$

Consequently, T is strongly monotone.

(D) T is is coercive,

we have for $u\in X$ with $\parallel u\parallel>1,$

\begin{align*}
\frac{1}{\parallel u\parallel}\langle Tu,u\rangle&=
\frac{1}{\parallel u\parallel}\int_{\Omega}(\mid \nabla
u\mid^{p(x)}+\mid u\mid^{p(x)})dx-\int_{\Omega}f(x,u)udx\\
&\geq \frac{1}{\parallel u\parallel} [\parallel
u\parallel^{p^{-}}-2\mid f\mid_{p'(x)}\mid u\mid_{p(x)}\\
&\geq\frac{1}{\parallel u\parallel} [\parallel
u\parallel^{p^{-}}-C_1\parallel u\parallel]
\end{align*}

which yields the coercivity of $T.$  A standar argument works whether replace the operator L by K. The previous steps guarantees
the existence of solution of the previous problems (\ref{e0})-
(\ref{e3}).

For the uniqueness of weak solution for problems studied, suppose
that u and v be a weak solutions such that $u \neq v$. By the strng
monocity o T it follows that

$$ 0=\langle Tu-Tv,u-v\rangle\geq C_p \parallel u-v\parallel^p\geq 0. $$

Then  $u=v$ and the proof now is completed.

Since $f(x,0)\neq 0,g(x,0)\neq 0$ then $T0\neq0.$

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\end{document}
