\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2009(2009), No. 144, pp. 1--11.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2009 Texas State University - San Marcos.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2009/144\hfil Weak solutions for anisotropic equations] {Weak solutions for anisotropic nonlinear elliptic equations with variable exponents} \author[B. Kone, S. Ouaro, S. Traore \hfil EJDE-2009/144\hfilneg] {Blaise Kone, Stanislas Ouaro, Sado Traore} % in alphabetical order \address{Blaise Kone \newline Laboratoire d'Analyse Math\'ematique des Equations (LAME)\\ Institut Burkinab\'e des Arts et M\'etiers, Universit\'e de Ouagadougou \\ 03 BP 7021 Ouaga 03 \\ Ouagadougou, Burkina Faso} \email{leizon71@yahoo.fr} \address{Stanislas Ouaro \newline Laboratoire d'Analyse Math\'ematique des Equations (LAME)\\ UFR. Sciences Exactes et Appliqu\'ees, Universit\'e de Ouagadougou \\ 03 BP 7021 Ouaga 03 \\ Ouagadougou, Burkina Faso} \email{souaro@univ-ouaga.bf, ouaro@yahoo.fr} \address{Sado Traore \newline Laboratoire d'Analyse Math\'ematique des Equations (LAME)\\ Institut des Sciences Exactes et Appliqu\'ees, Universit\'e de Bobo Dioulasso\\ 01 BP 1091 Bobo-Dioulasso 01 \\ Bobo Dioulasso, Burkina Faso} \email{sado@univ-ouaga.bf} \thanks{Submitted February 10, 2008. Published November 12, 2009.} \subjclass[2000]{35J20, 35J25, 35D30, 35B38, 35J60} \keywords{Anisotropic Sobolev spaces; weak energy solution; variable exponents; \hfill\break\indent electrorheological fluids} \begin{abstract} We study the anisotropic boundary-value problem \begin{gather*} -\sum^{N}_{i=1}\frac{\partial}{\partial x_{i}}a_{i}(x,\frac{\partial}{\partial x_{i}}u)=f \quad \text{in } \Omega, \\ u=0 \quad\text{on }\partial \Omega, \end{gather*} where $\Omega$ is a smooth bounded domain in $\mathbb{R}^{N}$ $(N\geq 3)$. We obtain the existence and uniqueness of a weak energy solution for $f\in L^{\infty}(\Omega)$, and the existence of weak energy solution for general data $f$ dependent on $u$. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{definition}[theorem]{Definition} \section{Introduction} Let $\Omega$ be a bounded domain of $\mathbb{R}^{N}$ $(N\geq 3)$ with smooth boundary $\partial \Omega$. Our aim is to prove existence and uniqueness of a weak energy solution to the anisotropic nonlinear elliptic problem \begin{equation} \label{e1.1} \begin{gathered} -\sum^{N}_{i=1}\frac{\partial}{\partial x_{i}} a_{i}(x,\frac{\partial}{\partial x_{i}}u)=f \quad \text{in } \Omega\\ u=0 \quad \text{on }\partial \Omega, \end{gathered} \end{equation} where the right hand side $f$ is in $L^{\infty}(\Omega)$. We assume that for $i=1,\dots ,N$ the function $a_{i}:\Omega\times\mathbb{R}\to\mathbb{R}$ is Carath\'eodory; i.e., $a(x,.)$ is continuous for a.e. $x\in\Omega$ and $a(.,t)$ is measurable for every $t\in \mathbb{R}$ and satisfy the following conditions: $a_{i}(x,\xi)$ is the continuous derivative with respect to $\xi$ of the mapping $A_{i}: \Omega\times\mathbb{R}\to\mathbb{R}$, $A_{i}=A_{i}(x,\xi)$; i.e., $a_{i}(x,\xi)=\frac{\partial}{\partial \xi}A_{i}(x,\xi)$ such that: The following equatility holds \begin{equation} \label{e1.2} A_{i}(x,0)=0, \end{equation} for almost every $x\in\Omega$. There exists a positive constant $C_{1}$ such that \begin{equation} \label{e1.3} |a_{i}(x,\xi)|\leq C_{1}(j_{i}(x)+|\xi|^{p_{i}(x)-1}) \end{equation} for almost every $x\in\Omega$ and for every $\xi\in \mathbb{R}$, where $j_{i}$ is a nonnegative function in $L^{p_{i}'(.)}(\Omega)$, with $1/p_{i}(x)+1/p_{i}'(x)=1$. The following inequality holds \begin{equation} \label{e1.4} \big(a_{i}(x,\xi)-a_{i}(x,\eta)\big).\big(\xi-\eta\big)>0 \end{equation} for almost every $x\in\Omega$ and for every $\xi,\eta\in\mathbb{R}$, with $\xi\neq\eta$. The following inequalities hold \begin{equation} \label{e1.5} |\xi|^{p_{i}(x)}\leq a_{i}(x,\xi).\xi\leq p_{i}(x)A_{i}(x,\xi) \end{equation} for almost every $x\in \Omega$ and for every $\xi\in\mathbb{R}$. For the exponent $p_{1}(.),\dots ,p_{N}(.)$, we assume that $p_{i}(.):\overline{\Omega}\to\mathbb{R}$ are continuous functions such that: \begin{equation} \label{e1.6} 2\leq p_{i}(x)1, \end{equation} where \[ p_{i}^{-}:=\mathop{\rm ess\,inf}_{x\in\Omega}p_{i}(x), p_{i}^{+}:=\mathop{\rm ess\,sup}_{x\in\Omega}p_{i}(x). \] A prototype example that is covered by our assumptions is the following anisotropic $(p_{1}(.),\dots ,p_{N}(.))$-harmonic problem: Set \[ A_{i}(x,\xi)=\big(1/p_{i}(x)\big)|\xi|^{p_{i}(x)}, \quad a_{i}(x,\xi)=|\xi|^{p_{i}(x)-2}\xi \] where $p_{i}(x)\geq 2$. Then we obtain the problem: \[ -\sum^{N}_{i=1}\frac{\partial}{\partial x_{i}} \Big(|\frac{\partial}{\partial x_{i}}u|^{p_{i}(x)-2} \frac{\partial}{\partial x_{i}}u\Big)=f \] which, in the particular case when $p_{i}=p$ for any $i\in\{1,\dots ,N\}$ is the $p(.)$-Laplace equation. The study of nonlinear elliptic equations involving the $p$-Laplace operator is based on the theory of standard Sobolev spaces $W^{m,p}(\Omega)$ in order to find weak solutions. For the nonhomogeneous $p(.)$-Laplace operators, the natural setting for this approach is the use of the variable exponent Lebesgue and Sobolev spaces $L^{p(.)}(\Omega)$ and $W^{m,p(.)}(\Omega)$. The spaces $L^{p(.)}(\Omega)$ and $W^{m,p(.)}(\Omega)$ were thoroughly studied by Musielak \cite{m3}, Edmunds et al \cite{e1,e2,e3}, Kovacik and Rakosnik \cite{k1}, Diening \cite{d1,d2} and the references therein. Variable Sobolev spaces have been used in the last decades to model various phenomena. Chen, Levine and Rao \cite{c1} proposed a framework for image restoration based on a variable exponent Laplacian. An other application which uses nonhomogeneous Laplace operators is related to the modelling of electrorheological fluids. The first major discovery in electrorheological fluids is due to Willis Winslow in 1949. These fluids have the interesting property that their viscosity depends on the electric field in the fluid. They can raise the viscosity by as much as five orders of magnitude. This phenomenon is known as the Winslow effect. For some technical applications, consult Pfeiffer et al \cite{p1}. Electrorheological fluids have been used in robotics and space technology. The experimental research has been done mainly in the USA, for instance in NASA Laboratories. For more information on properties, modelling and the application of variable exponent spaces to these fluids, we refer to Diening \cite{d1}, Rajagopal and Ruzicka \cite{r1}, and Ruzicka \cite{r2}. In this paper, the operator involved in \eqref{e1.1} is more general than the $p(.)$-Laplace operator. Thus, the variable exponent Sobolev space $W^{1,p(.)}(\Omega)$ is not adequate to study nonlinear problems of this type. This lead us to seek weak solutions for problems \eqref{e1.1} in a more general variable exponent Sobolev space which was introduced for the first time by Miha\"ilescu et al \cite{m1}. Note that, Antontsev and Shmarev \cite{a2} studied the following problem which is quite close to \eqref{e1.1}: \begin{equation} \label{e1.7} \begin{gathered} -\sum_{i}D_{i}(a_{i}(x,u))|D_{i}u|^{p_{i}(x)-2}D_{i}u+c(x,u) |u|^{\sigma(x)-2}u=F(x) \quad \text{in }\Omega\\ u=0 \quad \text{on }\partial\Omega, \end{gathered} \end{equation} in a bounded domain $\Omega\in\mathbb{R}^{N}$, and elliptic systems of the same structure, \begin{equation} \label{e1.8} \begin{gathered} -\sum_{j}D_{j}(a_{ij}(x,\nabla u))=f^{i}(x,u) \quad\text{ in }\Omega, \; i=1,\dots ,n.\\ u=0 \quad \text{on }\partial\Omega. \end{gathered} \end{equation} In \cite{a2}, the authors proved among others result, existence of (bounded) weak solutions and establish sufficient conditions of uniqueness of a weak solution, where the variational set considered is \[ \textbf{V}(\Omega)=\{u\in L^{\sigma(x)}(\Omega)\cap W^{1,1}_{0}(\Omega), D_{i}(u)\in L^{p_{i}(x)}(\Omega), i=1,\dots ,n\} \] equipped with the norm $\|u\|_{V}=\|u\|_{\sigma(.)} +\sum^{n}_{i=1}\|D_{i}u\|_{p_{i}(.)}$. The remaining part of this paper is organized as follows: Section 2 is devoted to mathematical preliminaries including, among other things, a brief discussion of variable exponent Lebesgue, Sobolev and anisotropic Sobolev variables exponent spaces. The main existence and uniqueness result is stated and proved in section 3. Finally, in section 4, we discuss some extensions. \section{Preliminaries} In this section, we define the Lebesgue and Sobolev spaces with variable exponent and give some of their properties. Roughly speaking, anistropic Lebesgue and Sobolev spaces are functional spaces of Lebesgue's and Sobolev's type in which different space directions have different roles. Given a measurable function $p(.):\Omega\to[1,\infty)$. We define the Lebesgue space with variable exponent $L^{p(.)}(\Omega)$ as the set of all measurable function $u:\Omega\to\mathbb{R}$ for which the convex modular \[ \rho_{p(.)}(u):=\int_{\Omega}|u|^{p(x)}dx \] is finite. If the exponent is bounded; i.e., if $p_{+}<\infty$, then the expression \[ |u|_{p(.)}:=\inf\{\lambda>0: \rho_{p(.)}(u/\lambda)\leq 1\} \] defines a norm in $L^{p(.)}(\Omega)$, called the Luxembourg norm. The space $(L^{p(.)}(\Omega),|.|_{p(.)})$ is a separable Banach space. Moreover, if $p_{-}>1$, then $L^{p(.)}(\Omega)$ is uniformly convex, hence reflexive, and its dual space is isomorphic to $L^{p'(.)}(\Omega)$, where $\frac{1}{p(x)}+\frac{1}{p'(x)}=1$. Finally, we have the H\"older type inequality: \begin{equation} \label{e2.1} \big|\int_{\Omega}uv\,dx\big|\leq\Big(\frac{1}{p_{-}}+\frac{1}{p'_{-}} \Big)|u|_{p(.)}|v|_{p'(.)}, \end{equation} for all $u\in L^{p(.)}(\Omega)$ and $v\in L^{p'(.)}(\Omega)$. Now, let \[ W^{1,p(.)}(\Omega):=\{u\in L^{p(.)}(\Omega):|\nabla u|\in L^{p(.)} (\Omega)\}, \] which is a Banach space equipped with the norm \[ \|u\|_{1,p(.)}:=|u|_{p(.)}+|\nabla u|_{p(.)}. \] An important role in manipulating the generalized Lebesgue-Sobolev spaces is played by the modular $\rho_{p(.)}$ of the space $L^{p(.)}(\Omega)$. We have the following result (cf. \cite{f1}). \begin{lemma} \label{lem2.1} If $u_{n}, u\in L^{p(.)}(\Omega)$ and $p_{+}<+\infty$ then the following relations hold \begin{itemize} \item[(i)] $|u|_{p(.)}>1\Rightarrow|u|_{p(.)}^{p_{-}} \leq \rho_{p(.)}(u)\leq |u|_{p(.)}^{p_{+}}$; \item[(ii)] $|u|_{p(.)}<1\Rightarrow|u|_{p(.)}^{p_{+}} \leq \rho_{p(.)}(u)\leq |u|_{p(.)}^{p_{-}}$; \item[(iii)] $|u_{n}-u|_{p(.)}\to 0\Rightarrow\rho_{p(.)}(u_{n}-u)\to 0$; \item[(iv)] $|u|_{L^{p(.)}(\Omega)}<1$ (respectively $=1;>1$) $\Leftrightarrow \rho_{p(.)}(u)<1$ (respectively $=1;>1$); \item[(v)] $|u_{n}|_{L^{p(.)}(\Omega)}\to 0$ (respectively $\to+\infty$) $\Leftrightarrow\rho_{p(.)}(u_{n})\to 0$ (respectively $\to+\infty$); \item[(vi)]$\rho_{p(.)}\big(u/|u|_{L^{p(.)}(\Omega)}\big)=1$. \end{itemize} \end{lemma} Next, we define $W^{1,p(.)}_{0}(\Omega)$ as the closure of $C_{0}^{\infty}(\Omega)$ in $W^{1,p(.)}(\Omega)$ under the norm $\|u\|_{1,p(.)}$. Set \[ C_{+}(\overline{\Omega})=\{p\in C(\overline{\Omega}): \min_{x\in \overline{\Omega}}p(x)>1\}. \] Furthermore, if $p\in C_{+}(\overline{\Omega})$ is logarithmic H\"older continuous, then $C_{0}^{\infty}(\Omega)$ is dense in $W^{1,p(.)}_{0}(\Omega)$, that is $H^{1,p(.)}_{0}(\Omega)=W^{1,p(.)}_{0}(\Omega)$ (cf. \cite{h1}). Since $\Omega$ is an open bounded set and $p\in C_{+}(\overline{\Omega})$ is logarithmic H\"older, the $p(.)$-Poincar\'e inequality \[ |u|_{p}\leq C|\nabla u|_{p(.)} \] holds for all $u\in W^{1,p(.)}_{0}(\Omega)$, where $C$ depends on $p$, $|\Omega|$, diam$(\Omega)$ and $N$ (see \cite{h1}), and so \[ \|u\|:=|\nabla u|_{p(.)}, \] is an equivalent norm in $W^{1,p(.)}_{0}(\Omega)$. Of course also the norm \[ \|u\|_{p(.)}:=\sum_{i=1}^{N}\big|\frac{\partial}{\partial x_{i}}u\big|_{p(.)} \] is an equivalent norm in $W^{1,p(.)}_{0}(\Omega)$. Hence the space $W^{1,p(.)}_{0}(\Omega)$ is a separable and reflexive Banach space. Finally, let us present a natural generalization of the variable exponent Sobolev space $W^{1,p(.)}_{0}(\Omega)$ (cf. \cite{m1}) that will enable us to study with sufficient accuracy problem \eqref{e1.1}. First of all, we denote by $\overrightarrow{p}:\overline{\Omega}\to \mathbb{R}^{N}$ the vectorial function $\overrightarrow{p}=(p_{1},\dots ,p_{N})$. The \emph{anisotropic variable exponent Sobolev space} $W^{1,\overrightarrow{p}(.)}_{0}(\Omega)$ is defined as the closure of $C^{\infty}_{0}(\Omega)$ with respect to the norm \[ \|u\|_{\overrightarrow{p}(.)}:=\sum_{i=1}^{N} \big|\frac{\partial}{\partial x_{i}}u\big|_{p_{i}(.)}. \] The space $(W^{1,\overrightarrow{p}(.)}_{0}(\Omega), \|u\|_{\overrightarrow{p}(.)})$ is a reflexive Banach space (cf. \cite{m1}). Let us introduce the following notation. \begin{gather*} \overrightarrow{P}_{+}=(p_{1}^{+},\dots ,p_{N}^{+}), \quad \overrightarrow{P}_{-}=(p_{1}^{-},\dots ,p_{N}^{-}),\\ P_{+}^{+}=\max\{p_{1}^{+},\dots ,p_{N}^{+}\}, \quad P_{-}^{+}=\max\{p_{1}^{-},\dots ,p_{N}^{-}\},\\ P_{-}^{-}=\min\{p_{1}^{-},\dots ,p_{N}^{-}\},\quad P_{-,\infty}=\max\{P_{-}^{+},P_{-}^{\ast}\}, \\ P_{-}^{\ast}=\frac{N}{\sum_{i=1}^{N}\frac{1}{p_{i}^{-}}-1}\,. \end{gather*} We have the following result (cf. \cite{m1}). \begin{theorem} \label{thm2.2} Assume $\Omega\subset\mathbb{R}^{N}$ $(N\geq 3)$ is a bounded domain with smooth boundary. Assume relation \eqref{e1.6} is fulfilled. For any $q\in C(\overline{\Omega})$ verifying \[ 10$. Since $\Lambda_{i}$ is convex (by Remark 2.3), we deduce that for any $v\in E$, the following inequality holds \[ \int_{\Omega}A_{i}(x,\frac{\partial}{\partial x_{i}} v)dx \geq\int_{\Omega}A_{i}(x,\frac{\partial}{\partial x_{i}} u)dx +\int_{\Omega}a_{i}(x,\frac{\partial}{\partial x_{i}} u). (\frac{\partial}{\partial x_{i}} v-\frac{\partial}{\partial x_{i}} u)dx. \] Using \eqref{e1.3} and \eqref{e2.1}, we have \begin{align*} \int_{\Omega}A_{i}(x,\frac{\partial}{\partial x_{i}} v)dx &\geq\int_{\Omega}A_{i}(x,\frac{\partial}{\partial x_{i}} u)dx -\int_{\Omega}|a_{i}(x,\frac{\partial}{\partial x_{i}} u)| |\frac{\partial}{\partial x_{i}} v-\frac{\partial}{\partial x_{i}} u|dx \\ &\geq\int_{\Omega}A_{i}(x,\frac{\partial}{\partial x_{i}} u)dx -C_{1}\int_{\Omega}j_{i}(x)|\frac{\partial}{\partial x_{i}} (v-u)|dx \\ &\quad -C_{1}\int_{\Omega}|\frac{\partial}{\partial x_{i}} u|^{p_{i}(x)-1} |\frac{\partial}{\partial x_{i}} (v-u)|dx \\ &\geq\int_{\Omega}A_{i}(x,\frac{\partial}{\partial x_{i}}u)dx -C_{2}|j_{i}|_{p_{i}'(x)}|\frac{\partial}{\partial x_{i}} (v-u)|_{p_{i}(x)} \\ &\quad -C_{3}||\frac{\partial}{\partial x_{i}} u|^{p_{i}(x)-1}|_{p_{i}'(x)}|\frac{\partial}{\partial x_{i}} (v-u)|_{p_{i}(x)} \\ &\geq\int_{\Omega}A_{i}(x,\frac{\partial}{\partial x_{i}} u)dx -C_{4}\|v-u\|_{\overrightarrow{p}(.)}\\ &\geq\int_{\Omega} A_{i}(x,\frac{\partial}{\partial x_{i}} u)dx-\epsilon, \end{align*} for all $v\in E$ with $\|v-u\|_{\overrightarrow{p}(.)}<\delta=\epsilon/C_{4}$, where $C_{2}, C_{3}$ and $C_{4}$ are positive constants. We conclude that $\Lambda_{i}$ is weakly lower semi-continuous for $i=1,\dots ,N$. The proof is complete. \end{proof} \begin{proposition} \label{prop3.6} The functional $J$ is bounded from below, coercive and weakly lower semi-continuous. \end{proposition}. \begin{proof} Using \eqref{e1.5}, we have \begin{align*} J(u)&=\int_{\Omega}\sum_{i=1}^{N}A_{i}(x, \frac{\partial}{\partial x_{i}} u)dx -\int_{\Omega}fu\,dx\\ &\geq \frac{1}{P_{+}^{+}} \sum_{i=1}^{N}\int_{\Omega}|\frac{\partial}{\partial x_{i}} u|^{p_{i}(x)} dx-\int_{\Omega}fu\,dx \\ &\geq\frac{1}{P_{+}^{+}}\sum_{i=1}^{N} \int_{\Omega}|\frac{\partial}{\partial x_{i}} u|^{p_{i}(x)}dx -\|f\|_{q'}\|u\|_{q}, \end{align*} where $\|u\|_{q}=\big(\int_{\Omega}|u|^{q}dx\big)^{1/q}$ and $11. \end{cases} \] For the coerciveness of $J$, we focus our attention on the case when $u\in E$ and $\|u\|_{\overrightarrow{p}(.)}>1$. Then, by Lemma 2.1 we obtain \begin{align*} J(u) &\geq\frac{1}{P_{+}^{+}}\sum_{i=1}^{N}| \frac{\partial}{\partial x_{i}} u|_{p_{i}(.)}^{\alpha_{i}} -\|f\|_{q'}\|u\|_{q}\\ &\geq\frac{1}{P_{+}^{+}}\sum_{i=1}^{N}|\frac{\partial}{\partial x_{i}} u|_{p_{i}(.)}^{P_{-}^{-}}-\frac{1}{P_{+}^{+}} \sum_{\{i:\alpha_{i}=P^{+}_{+}\}} \Big(|\frac{\partial}{\partial x_{i}} u|_{p_{i}(.)}^{P^{-}_{-}} -|\frac{\partial}{\partial x_{i}} u|_{p_{i}(.)}^{P^{+}_{+}}\Big) -\|f\|_{q'}\|u\|_{q}\\ &\geq\frac{1}{P_{+}^{+}}\sum_{i=1}^{N}| \frac{\partial}{\partial x_{i}} u|_{p_{i}(.)}^{P_{-}^{-}} -\frac{1}{P_{+}^{+}}\sum_{\{i:\alpha_{i}=P^{+}_{+}\}} \Big(|\frac{\partial}{\partial x_{i}} u|_{p_{i}(.)}^{P^{-}_{-}}\Big) -\|f\|_{q'}\|u\|_{q}\\ &\geq\frac{1}{P_{+}^{+}} \sum_{i=1}^{N}|\frac{\partial}{\partial x_{i}} u|_{p_{i}(.)}^{P_{-}^{-}} -\frac{N}{P_{+}^{+}}-\|f\|_{q'}\|u\|_{q}\\ &\geq \frac{1}{P_{+}^{+}}\sum_{i=1}^{N}|\frac{\partial}{\partial x_{i}} u|_{p_{i}(.)}^{P_{-}^{-}}-\frac{N}{P_{+}^{+}}-C'\|u\|_{q}\\ &\geq\frac{1}{P_{+}^{+}}\Big(\frac{1}{N} \sum_{i=1}^{N}|\frac{\partial}{\partial x_{i}} u|_{p_{i}(.)} \Big)^{P^{-}_{-}} -\frac{N}{P_{+}^{+}}-C'\|u\|_{q}\\ &\geq\frac{1}{P_{+}^{+} N^{P_{-}^{-}}}\|u\|_{\overrightarrow{p}(.)}^{P_{-}^{-}} -\frac{N}{P_{+}^{+}}-C'\|u\|_{\overrightarrow{p}(.)}, \end{align*} since $E$ is continuously embedded in $L^{q}(\Omega)$. As $P_{-}^{-}>1$, then $J$ is coercive. It is obvious that $J$ is bounded from below. By Lemma 3.5, $\Lambda_{i}$ is weakly lower semi-continuous for $i=1,\dots ,N$. We show that $J$ is weakly lower semi-continuous. Let $(u_{n})\subset E$ be a sequence which converges weakly to $u$ in $E$. Since for $i=1,\dots ,N$ $\Lambda_ {i}$ is weakly lower semi-continuous, we have \begin{equation} \label{e3.2} \Lambda_{i}(u)\leq \liminf_{n\to+\infty}\Lambda_{i}(u_{n}). \end{equation} On the other hand, $E$ is embedded in $L^{q}(\Omega)$ for $10$, $C_{2}>0$ such that \begin{equation} \label{e4.2} |f(x,t)|\leq C_{1}+C_{2}|t|^{\beta-1}, \end{equation} where $1<\beta1$. We know that $E$ is continuously embedded in $L^{\beta}(\Omega)$. It follows from inequality above that \[ J(u)\geq C_{5}\|u\|_{\overrightarrow{p}(.)}^{P_{-}^{-}} -\frac{N}{P^{+}_{+}}-C_{4}\|u\|^{\beta}_{\overrightarrow{p}(.)} -C_{3}\to +\infty \] as $\|u\|_{\overrightarrow{p}(.)}\to+\infty$. Consequently, $J$ is bounded from below and coercive. The proof is then complete. \end{proof} Assume now that $F^{+}(x,t)=\int_{0}^{t}f^{+}(x,s)ds$ is such that there exists $C_{1}>0$, $C_{2}>0$ such that \begin{equation} \label{e4.3} |f^{+}(x,t)|\leq C_{1}+C_{2}|t|^{\beta-1}, \end{equation} where $1<\beta