\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2011 (2011), No. 62, pp. 1--16.\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{9mm}} \begin{document} \title[\hfilneg EJDE-2011/62\hfil Quasilinear elliptic problems] {Quasilinear elliptic problems with nonstandard growth} \author[M. B. Benboubker, E. Azroul, A. Barbara \hfil EJDE-2011/62\hfilneg] {Mohamed Badr Benboubker, Elhoussine Azroul, Abdelkrim Barbara} \address{Mohamed Badr Benboubker \newline University of Fez, Faculty of Sciences Dhar El Mahraz, Laboratory LAMA, Department of Mathematics, B.P 1796 Atlas Fez, Morocco} \email{simo.ben@hotmail.com} \address{Elhoussine Azroul \newline University of Fez, Faculty of Sciences Dhar El Mahraz, Laboratory LAMA, Department of Mathematics, B.P 1796 Atlas Fez, Morocco} \email{azroul\_elhoussine@yahoo.fr} \address{Abdelkrim Barbara \newline University of Fez, Faculty of Sciences Dhar El Mahraz, Laboratory LAMA, Department of Mathematics, B.P 1796 Atlas Fez, Morocco} \email{abdelkrim.barbara@yahoo.com} \thanks{Submitted November 1, 2010. Published May 11, 2011.} \subjclass[2000]{35J20, 35J25} \keywords{Quasilinear elliptic equation; Sobolev spaces with variable exponent; \hfill\break\indent image processing; Dirichlet problem} \begin{abstract} We prove the existence of solutions to Dirichlet problems associated with the $p(x)$-quasilinear elliptic equation $$ Au =- \operatorname{div} a(x,u,\nabla u)= f(x,u,\nabla u). $$ These solutions are obtained in Sobolev spaces with variable exponents. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} Partial differential equations with non-standard growth in Lebesgue and Sobolev spaces with variable exponent have been a very active field of investigation in recent years. The present line of investigation goes back to an article by Kov\'a\u(c)ik and R\'akosnik \cite{Kov&Ras} in 1991. The development, mainly by Ru\u{z}i\u{c}ka \cite{Ruz}, of a theory modelling the behavior of electro-rhelogical fluid, an important class of non-Newtonian fluids, seems to have boosted a still far from completed effort to study and understand nonlinear PDEs involving variable exponents by several researches. Samko \cite{sam1,sam2,sam3} working based on earlier Russian work (Sharapudinov \cite{shara} and Zhikov \cite{zhikov}), Fan and collaborators \cite{fan1,fan2,fan3,fan4} drawing inspiration from the study of differential equations(e.g. Marcellini \cite{marc}). More recently, an application to image processing was proposed by Chen, Levine and Rao \cite{Rao}. To give the reader a feeling for the idea behind this application we mention that the proposed model requires the minimization over u of the energy, \begin{equation} E(u) =\int_{\Omega}|\nabla u(x)|^{p(x)}+|u(x)-I(x)|^{2}dx, \label{eq.im1} \end{equation} where $I$ is a given input. Recall that in the constant exponent case, the power $p=2$ corresponds to isotropic smoothing, which corresponds to minimizing the energy, \begin{equation} E_2(u)=\int_{\Omega}|\nabla u(x)|^2+|u(x)-I(x)|^2dx. \end{equation} Unfortunately, the smoothing will destroy all small details from the image, so this procedure is not very useful. Where as $p=1$ gives total variations smoothing which corresponds to minimizing the energy, \begin{equation} E_1(u)=\int_{\Omega}|\nabla u(x)|+|u(x)-I(x)|^2dx. \end{equation} The benefit of this approach not only preserves edges, it also creates edges where there were none in the original image (the so-called staircase effect). As the strengths and weaknesses of these two methods for image restoration are opposite, it is a natural to try to combine them. That was the idea of Chen, Levine and Rao \cite{Rao}, looking at $E_1$ and $E_2$ suggests that the appropriate energy is $E(u)$ (see \ref{eq.im1}), where $p(x)$, is a function varying between 1 and 2. This function should be close to 1 where there are likely to be edges, and close to 2 where there are likely not to be edges, and depends on the location , $x$, in the image. In this way the direction and speed of diffusion at each location depends on the local behavior. We point out that, this model is linked with energy which can be associated to the $p(x)$-Laplacian operators; i.e., \begin{equation} \Delta_{p(x)}u=\operatorname{div} (|\nabla u|^{p(x)-2}\nabla u). \end{equation} Moreover, the choice of the exponent yields a variational problem which has an Euler-Lagrange equation, and the solution can be found by solving corresponding evolutionary PDE. In this paper, we consider a problem with potential applications. This problem has already been treated for constant exponent but it seems to be more realistic to assume the exponent to be variable. More precisely, we are interested in this paper to the following Dirichlet problems \begin{equation} \begin{gathered} Au = f(x,u,\nabla u) \quad \text{in } D'(\Omega), \\ u = 0 \quad \text{on } \partial\Omega, \end{gathered} \label{eq1} \end{equation} where $\Omega$ is a bounded open subset of $\mathbb{R}^N$ $(N\geq 2)$, and $p\in {\mathcal{C}}(\bar{\Omega})$, $p(x) > 1$, and where $A$ is a Leray-Lions operator defined from $W_0^{1,p(x)}(\Omega)$ into its dual $W^{-1,p'(x)}(\Omega)$ by the formula \begin{equation} Au = - \operatorname{div}a(x,u,\nabla u) \end{equation} and where $f:\Omega\times \mathbb{R} \times \mathbb{R}^N \to \mathbb{R} $ is a Carath\'{e}odory function which satisfies the growth condition \begin{equation} |f(x,r,\xi)| \leq g(x)+ |r|^{\eta(x)}+|\xi|^{\delta(x)},\label{eq2} \end{equation} where $0\leq \eta(x) < p(x)-1 $ and $ 0 \leq \delta(x) < \big(p(x)-1\big)/p'(x)$. In the case of non-variables exponents, Boccardo, Murat and Puel have studied in \cite{bomu} the problem \eqref{eq1} with $f$ satisfying the condition \begin{equation} |f(x,r,\xi)| \leq h(|r|)(1+|\xi|^{p}),\label{eq3} \end{equation} where $h$ is an increasing function from $\mathbb{R} ^{+} \to \mathbb{R} ^{+}$. Kuo and Tsai \cite{Kao}, proved the existence results under the growth condition \begin{equation}|f(x,r,\xi)| \leq C (1+|r|^{\delta}+|\xi|^{p}).\label{eq4} \end{equation} However, in the case of variable exponent, we can list the work of Fan and Zhang \cite{Fan} who studied the particular case \begin{equation} \begin{gathered} - \operatorname{div}(|\nabla u|^{p(x)-2} \nabla u) = f(x,u) \quad x \in\Omega\\ u = 0 \quad\text{on } \partial \Omega ,\\ \end{gathered} \label{eq5} \end{equation} where $f$ satisfies the growth condition \begin{equation} |f(x,r)|\leq C_1+C_2|r|^{\beta(x)-1}, \end{equation} with $1\leq \beta 1 \text{ for any } x\in \bar \Omega\}, $$ and we define the variable exponent Lebesque space by: $$ L^{p(x)}(\Omega) = \{u \text{ is a measurable real-valued function, } \int_{\Omega} |u(x)|^{p(x)} dx < \infty\}, $$ We can introduce the norm on $L^{p(x)}(\Omega)$ by $$ |u|_{p(x)} = \inf \big\{\lambda > 0, \int_{\Omega} |\frac{u(x)}{\lambda}|^{p(x)} \leq 1 \big\}. $$ \begin{remark} \label{rmk2.1} \rm Note that the variable exponent Lebesgue spaces resemble classical Lebesgue spaces in many respects: they are Banach spaces (Kov\'a\u(c)ik and R\'akosnik \cite[Theorem 2.5]{Kov&Ras}), the H\"{o}lder inequality holds (Kov\'a\u(c)ik and R\'akosnik \cite[Theorem 2.1]{Kov&Ras}), they are reflexive if and only if $ 1 1$) if and only if $\rho(u) < 1$ (resp. $= 1$, $> 1$); \item[(ii)] $|u|_{p(x)} > 1$ implies $|u|_{p(x)}^{p^{-}} \leq \rho(u) \leq |u|_{p(x)}^{p^{+}}$; $|u|_{p(x)} < 1$ implies $|u|_{p(x)}^{p^{+}} \leq \rho(u) \leq |u|_{p(x)}^{p^{-}}$; \item[(iii)] $ |u|_{p(x)} \to 0$ if and only if $\rho(u) \to 0$; $|u|_{p(x)} \to \infty$ if and only if $\rho(u) \to \infty$. \end{itemize} \end{proposition} We define the variable Sobolev space by $$ W^{1,p(x)}(\Omega) = \{u\in L^{p(x)}(\Omega) : |\nabla u|\in L^{p(x)}(\Omega) \}. $$ with the norm \begin{equation}\label{norm} \|u\| = |u|_{p(x)} + |\nabla u|_{p(x)} \quad \forall u \in W ^{1,p(x)}(\Omega). \end{equation} We denote by $ W_0^{1,p(x)}(\Omega)$ the closure of $ C_0^{\infty}(\Omega)$ in $W^{1,p(x)}(\Omega)$ and $ p^{*}(x) = \frac{N p(x)}{N - p(x)}$, for $p(x) < N$. \begin{proposition}[\cite{Fan2}] \label{prop3} \begin{itemize} \item[(i)] Assuming $p^{-}> 1$, the spaces $W^{1, p(x)}(\Omega)$ and $W_0^{1, p(x)}(\Omega)$ are separable and reflexive Banach spaces. \item[(ii)] if $q\in {\mathcal{C}}_{+}(\bar{\Omega})$ and $q(x) < p^{*}(x) $ for any $x \in \overline{\Omega}$, then $ W^{1, p(x)}(\Omega) \hookrightarrow\hookrightarrow L^{q(x)}(\Omega)$ is compact and continuous. \item[(iii)] There is a positive constant $C$, such that $$ | u|_{p(x)} \leq C |\nabla u|_{p(x)} \quad \forall u\in W_0^{1,p(x)}(\Omega). $$ \end{itemize} \end{proposition} \begin{remark} \label{rem2.2} \rm By (iii) of Proposition \ref{prop3}, we know that $|\nabla u|_{p(x)}$ and $\|u\|$ are equivalent norms on $W^{1,p(x)}_0$. \end{remark} \section{Basic assumptions and some Lemmas} Let $p\in {\mathcal{C}}_{+}(\bar{\Omega})$ such that $ 1 < p^{-} \leq p(x) \leq p^{+} < N$, and denote $$ Au = - \operatorname{div}a(x,u,\nabla u), $$ where $a:\Omega \times \mathbb{R} \times \mathbb{R}^N \to \mathbb{R}^N$ is a carath\'eodory function satisfying the following assumptions: \begin{itemize} \item[(H1)] $|a(x,r,\xi)|\leq \beta [k(x) + |r|^{p(x) - 1} + |\xi|^{p(x) - 1}]$; %\label{ass1} \item[(H2)] $[a(x,r,\xi) - a(x,r,\eta)](\xi - \eta) > 0$ for all $\xi \neq \eta \in \mathbb{R}^N$; %, \label{ass2} \item[(H3)] $a(x,r,\xi)\xi \geq \alpha |\xi|^{p(x)}$; %\label{ass3} \end{itemize} for a.e. $x \in \Omega$, all $(r,\xi) \in \mathbb{R} \times \mathbb{R}^N$, where $k(x)$ is a positive function lying in $L^{p'(x)}(\Omega)$ and $ \beta,\alpha > 0$. Let $f$ be a Carath\'{e}odory function defined on $\Omega \times \mathbb{R} \times \mathbb{R}^N$ such that \begin{itemize} \item[(H4)] $|f(x,r,\xi)| \leq g(x) + |r|^{\eta(x)} + |\xi|^{\delta(x)}$ %\label{ass4} for a.e. $x \in \Omega$, all $(r,\xi) \in \mathbb{R} \times \mathbb{R}^N$, where $ g:\Omega \to \mathbb{R}^{+}$, $g\in L^{p'(x)}(\Omega) $ and $ 0 \leq \eta(x)< p(x) - 1$, $ 0 \leq \delta(x)<\frac{p(x) - 1}{p'(x)} $. \end{itemize} \begin{definition} \label{def} \rm Let $Y$ be a separable reflexive Banach space. An operator $B$ defined from $Y$ to its dual $Y^{*}$ is called an operator of the calculus of variations type, if $B$ is bounded and is of the form \begin{equation} B(u) = B(u,u), \end{equation} where $(u,v) \to B(u,v)$ is an operator defined from $Y \times Y$ into $Y^{*}$ which satisfying the following properties: \begin{equation} \label{e3.2} \parbox{10cm}{For $u \in Y$, the mapping $v \to B(u,v)$ is bounded hemicontinuous from $ Y$ into $Y^{*}$ and $(B(u,u)-B(u,v),u-v) \geq 0$;} \end{equation} for $v \in Y$, the mapping $u \to B(u,v)$ is bounded hemicontinous from $Y$ into $Y^{*}$; \begin{equation} \label{e3.3} \parbox{10cm}{if $u_{n} \rightharpoonup u$ in $Y$ and if $(B(u_{n},u_{n}) - B(u_{n},u),u_{n} - u ) \to 0$ , then $B(u_{n}, v) \rightharpoonup B(u,v)$ in $Y^{*}$ for all $v \in Y$.} \end{equation} and \begin{equation} \label{e3.4} \parbox{10cm}{if $u_{n} \rightharpoonup u$ in $Y$ and if $B(u_{n},v)\rightharpoonup \psi$ in $Y^{*}$, then $(B(u_{n}, v),u_{n}) \to (\psi,u)$.} \end{equation} The symbol $\rightharpoonup$ denote the weak convergence. \end{definition} \begin{lemma}\label{lem1} Assume that {\rm (H1)--(H4)} are satisfied and let $(u_{n})_{n}$ be a sequence in $W_0^{1,p(x)}(\Omega)$ and let $u \in W_0^{1,p(x)}(\Omega)$. If $u_{n} \rightharpoonup u$ in $W_0^{1,p(x)}(\Omega)$, then for some subsequence denoted again $(u_{n})$, we have $$ a(x, u_{n},\nabla v) \to a(x, u,\nabla v) \quad \text{in } \big( L^{p'(x)}(\Omega)\big)^N, \forall v \in W_0^{1,p(x)}(\Omega). $$ \end{lemma} \begin{proof} From (H1), it follows that \begin{equation}\label{eq.lem.1} \begin{aligned} & |a(x, u_{n}, \nabla v)|^{p'(x)}\\ & \leq \beta^{p'(x)}[k(x) + |u_{n}|^{p(x) - 1} + |\nabla v|^{p(x) - 1}]^{p'(x)} \\ & \leq (\beta+1)^{p'^{+}}2^{p'^{+} - 1} [ k(x)^{p'(x)} + 2^{p'^{+} - 1} ( |u_{n}|^{(p(x) - 1) p'(x)} + | \nabla v|^{(p(x) - 1) p'(x)} ) ] \\ & \leq (\beta+1)^{p'^{+}}2^{2 ( p'^{+} - 1)}[ k(x)^{p'(x)} + |u_{n}|^{p(x)} + |\nabla v|^{p(x)} ] . \end{aligned} \end{equation} In the second inequality above we have used \cite{sanchon}. Since $u_{n}\rightharpoonup u$ in $W_0^{1,p(x)}(\Omega)$ and according to proposition \ref{prop3}, we have $W_0^{1,p(x)}(\Omega) \hookrightarrow\hookrightarrow L^{p(x)}$ is compact and continuous, there exists a subsequence denoted again $(u_{n})$ such that, $u_{n} \to u$ in $L^{p(x)}(\Omega)$, and therefore a.e. in $\Omega$; hence \begin{equation} |a(x, u_{n},\nabla v)|^{p'(x)} \to |a(x, u,\nabla v)|^{p'(x)} \quad \text{a.e. in } \Omega, \end{equation} and \begin{equation} \label{equi0} \begin{split} &(\beta+1)^{p'^{+}}2^{2 ( p'^{+} - 1)}[ k(x)^{p'(x)} + |u_{n}|^{p(x)} + |\nabla v|^{p(x)} ] \\ & \to (\beta+1)^{p'^{+}}2^{2 ( p'^{+} - 1)}[k(x)^{p'(x)} + |u|^{p(x)} + |\nabla v|^{p(x)}] \text{ a.e. in } \Omega. \end{split} \end{equation} For each measurable subset $E$, we have \begin{equation}\label{equi} \begin{split} &\int_{E}|a(x, u_{n}, \nabla v)|^{p'(x)}dx\\ &\leq (\beta+1)^{p'^{+}}2^{2 ( p'^{+} - 1)} \Big[ \int_{E} k(x)^{p'(x)} dx + \int_{E} |u_{n}|^{p(x)} dx + \int_{E} |\nabla v|^{p(x)}dx \Big], \end{split} \end{equation} in view of \eqref{equi0} and \eqref{equi}, there exists $\eta(\varepsilon)$ such that $$ \int_{E}|a(x, u_{n}, \nabla v)|^{p'(x)}dx<\varepsilon $$ for all $E$ with $\operatorname{meas}(E)<\eta(\varepsilon)$, which implies the equi-integrability of $a(x, u_{n}, \nabla v)$. Finaly by Vitali's theorem, \begin{equation} a(x, u_{n},\nabla v) \to a(x, u,\nabla v)\quad \text{in } \big(L^{p'(x)}(\Omega)\big)^N. \end{equation} \end{proof} \begin{lemma}\label{lem2} Let $g\in L^{r(x)}(\Omega)$ and $g_{n}\in L^{r(x)}(\Omega)$ with $|g_{n}|_{L^{r(x)}(\Omega)} \leq C$ for $1< r(x)< \infty$. If $g_{n}(x)\to g(x)$ a.e. in $\Omega$, then $g_{n}\rightharpoonup g$ in $L^{r(x)}(\Omega)$. \end{lemma} \begin{proof} Let $$ E(N)=\{x \in \Omega: |g_{n}(x)-g(x)|\leq 1,\, \forall n\geq N\}. $$ Since $\operatorname{meas}(E(N)) \to \operatorname{meas}(\Omega)$ as $N\to \infty$, and setting $$ {\mathcal{F}}=\{\varphi_{{N}}\in L^{r'(x)}(\Omega): \varphi_{{N}}\equiv 0 \text{ a.e. in } \Omega\backslash E(N)\}, $$ we shall show that ${\mathcal{F}}$ is dense in $L^{r'(x)}(\Omega)$. Let $f\in L^{r'(x)}(\Omega)$, we set $$ f_{{N}}(x)=\begin{cases} f(x)&\text{if } x\in E(N),\\ 0&\text{if } x\in\Omega\backslash E(N). \end{cases} $$ Then \begin{align*} \rho_{r'(x)}(f_{{N}}-f) &= \int_{\Omega}|f_{{N}}(x)-f(x)|^{r'(x)}dx \\ &= \int_{E(N)}|f_{{N}}(x)-f(x)|^{r'(x)}dx +\int_{\Omega\backslash E(N)}|f_{{N}}(x)-f(x)|^{r'(x)}dx \\ &= \int_{\Omega\backslash E(N)}|f(x)|^{r'(x)}dx \\ &= \int_{\Omega}|f(x)|^{r'(x)}\chi_{\Omega\backslash E(N)}dx \end{align*} Taking $\psi_{{N}}(x)=|f(x)|^{r'(x)}\chi_{{\Omega\backslash E(N)}}$ for almost every $x$ in $\Omega$, we obtain $$ \psi_{{N}} \to 0 \text{ a.e. in }\Omega\quad\text{and}\quad |\psi_{{N}}|\leq |f|^{r'(x)}. $$ Using the dominated convergence theorem, we have $\rho_{r'(x)}(f_{{N}}-f) \to 0$ as $N\to\infty$; therefore $f_{{N}} \to f$ in $L^{r'(x)}(\Omega)$. Consequently ${\mathcal{F}}$ is dense in $L^{r'(x)}(\Omega)$. Now we shall show that \[ \lim_{n\to\infty}\int_{\Omega}\varphi(x)\big(g_{n}(x)-g(x)\big)dx=0, \quad \forall\ \varphi\in {\mathcal{F}}. \] Since $\varphi\equiv0$ in $\Omega\setminus E(N)$, it suffices to prove that $$ \int_{E(N)}\varphi(x)(g_{n}(x)-g(x))dx \to 0 \quad\text{as } n \to \infty. $$ We set $\phi_{n}=\varphi\big(g_{n}-g\big)$. Since $|\varphi(x)\|g_{n}(x)-g(x)| \leq |\varphi(x)|$ a.e. in $E(N)$ and $\phi_{n} \to 0$ a.e. in $\Omega$, thanks to the dominated convergence theorem, we deduce $\phi_{n} \to 0$ in $L^{1}(\Omega)$. Which implies that $$ \lim_{n\to\infty}\int_{\Omega}\varphi(x)(g_{n}(x)-g(x))dx=0,\quad \forall\ \varphi\in {\mathcal{F}} $$ Now, by the density of ${\mathcal{F}}$ in $L^{r'(x)}(\Omega)$, we conclude that $$ \lim_{n\to\infty}\int_{\Omega}\varphi g_{n}dx=\int_{\Omega}\varphi gdx, \quad \forall \varphi\in L^{r'(x)}(\Omega). $$ Finally $g_{n} \rightharpoonup g$ in $L^{r(x)}(\Omega)$. \end{proof} \begin{lemma}\label{lem3} Assume {\rm (H1)--(H4)}, and let $(u_{n})_{n}$ be a sequence in $W_0^{1,p(x)}(\Omega)$ such that $u_{n} \rightharpoonup u$ in $W_0^{1,p(x)}(\Omega)$ and \begin{equation}\label{eq.lem4.2} \int_{\Omega} [a(x, u_{n}, \nabla u_{n}) - a(x, u_{n}, \nabla u)]\nabla (u_{n} - u ) \,dx\to 0. \end{equation} Then, $u_{n}\to u$ in $W_0^{1,p(x)}(\Omega)$. \end{lemma} \begin{proof} Let $ D_{n} = [a(x,u_{n}, \nabla u_{n}) - a(x,u_{n}, \nabla u)] \nabla(u_{n} - u) $. Then by (H2), $D_{n}$ is a positive function, and by \eqref{eq.lem4.2} $ D_{n} \to 0$ in $L^{1}(\Omega)$. Extracting a subsequence, still denoted by $u_{n}$, we can write $u_{n} \rightharpoonup u$ in $W_0^{1,p(x)}(\Omega)$ which implies $u_{n} \to u$ a.e. in $\Omega$, Similarly $D_{n} \to 0$ a.e. in $\Omega$. Then there exists a subset $B$ of $\Omega$, of zero measure, such that for $x \in \Omega \setminus B$, $|u(x)|<\infty$, $|\nabla u(x)|<\infty$, $k(x)< \infty$, $u_{n}(x) \to u(x)$, $D_{n}(x) \to 0$. Defining $\xi_{n} = \nabla u_{n}(x)$, $\xi = \nabla u(x)$, we have \begin{equation} \label{eq.lem3.3} \begin{aligned} D_{n}(x) &=[a(x,u_{n}, \xi_{n}) - a(x,u_{n}, \xi)] (\xi_{n} - \xi) \\ &= a(x,u_{n}, \xi_{n}) \xi_{n} + a(x,u_{n}, \xi) \xi - a(x,u_{n}, \xi_{n}) \xi - a(x,u_{n}, \xi) \xi_{n} \\ &\geq \alpha |\xi_{n}|^{p(x)} + \alpha |\xi|^{p(x)} - \beta (k(x) + |u_{n}|^{p(x) - 1} + |\xi_{n}|^{p(x) - 1} )|\xi|\\ &\quad - \beta (k(x) + |u_{n}|^{p(x) - 1} + |\xi|^{p(x) - 1} ) |\xi_{n}|\\ &\geq \alpha |\xi_{n}|^{p(x)} - C_{x} \big[1 + |\xi_{n}|^{p(x) - 1} + |\xi_{n}| \big], \end{aligned} \end{equation} where $ C_{x} $ is a constant which depends on $x$, but does not depend on $n$. Since $u_{n}(x) \to u(x)$ we have $|u_{n}(x)| \leq M_{x}$, where $M_{x}$ is some positive constant. Then by a standard argument $|\xi_{n}|$ is bounded uniformly with respect to $n$, indeed \eqref{eq.lem3.3} becomes \begin{equation} D_{n}(x) \geq |\xi_{n}|^{p(x)} \big(\alpha - \frac{C_{x}}{|\xi_{n}|^{p(x)}} - \frac{C_{x}}{|\xi_{n}|} - \frac{C_{x}}{|\xi_{n}|^{p(x) - 1}}\big). \end{equation} If $|\xi_{n}| \to \infty$ (for a subsequence), then $D_{n}(x)\to \infty$ which gives a contradiction. Let now $\xi^{*}$ be a cluster point of $\xi_{n}$. We have $|\xi^{*}| < \infty$ and by the continuity of $a$ we obtain \begin{equation} [a(x,u(x), \xi^{*}) - a(x,u(x), \xi) ](\xi^{*} - \xi) = 0. \end{equation} In view of (H2), we have $\xi^{*}= \xi$. The uniqueness of the cluster point implies \begin{equation} \nabla u_{n}(x) \to \nabla u(x) \quad \text{a.e.in } \Omega. \end{equation} Since the sequence $a(x,u_{n},\nabla u_{n})$ is bounded in $(L^{p'(x)}(\Omega))^N$, and $a(x,u_{n},\nabla u_{n}) \to a(x,u,\nabla u)$ a.e. in $\Omega$, Lemma \ref{lem2} implies \begin{equation} a(x,u_{n},\nabla u_{n}) \rightharpoonup a(x,u,\nabla u) \quad \text{in } (L^{p'(x)}(\Omega))^N \; \text{a.e. in } \Omega. \end{equation} We set $\bar{y}_{n} = a(x, u_{n}, \nabla u_{n})\nabla u_{n}$ and $\bar{y} = a(x, u, \nabla u)\nabla u$. As in \cite{bomu} we can write $$ \bar{y}_{n} \to \bar{y} \text{in} L^{1}(\Omega). $$ By (H3) we have $$ \alpha |\nabla u_{n}|^{p(x)} \leq a(x, u_{n}, \nabla u_{n})\nabla u_{n}. $$ Let $ z_{n} = |\nabla u_{n}|^{p(x)}$, $z = |\nabla u|^{p(x)}$, $y_{n} = \frac{\bar{y}_{n}}{\alpha}$, and $y = \frac{\bar{y}}{\alpha}$. Then by Fatou's lemma, \begin{equation} \int_{\Omega} 2y \,dx \leq \liminf_{n\to \infty} \int_{\Omega} y + y_{n} - |z_{n} - z |\, dx; \end{equation} i.e., $ 0 \leq - \limsup\limits_{n\to \infty} \int_{\Omega} |z_{n} - z | dx$. Then \begin{equation} 0 \leq \liminf_{n\to \infty} \int_{\Omega} |z_{n} - z | dx \leq \limsup_{n\to \infty} \int_{\Omega} |z_{n} - z | dx \leq 0, \end{equation} this implies \begin{equation} \nabla u_{n} \to \nabla u \quad \text{in } (L^{p(x)}(\Omega))^N. \end{equation} Hence $u_{n} \to u$ in $ W_0^{1,p(x)}(\Omega)$, which completes the present proof. \end{proof} For $v \in W_0^{1,p(x)}(\Omega)$, we associate the Nemytskii operator $F$ with respect to $f$, defined by \begin{equation} F(v, \nabla v)(x) = f(x,v,\nabla v)) \quad \text{a.e. $x$ in } \Omega . \end{equation} \begin{lemma}\label{lem4} The mapping $v \mapsto F(v,\nabla v) $ is continuous from the space $W_0^{1, p(x)}(\Omega)$ to the space $L^{p'(x)}(\Omega)$. \end{lemma} \begin{proof} By (H4), we have \begin{equation} |f(x,r,\xi)| \leq g(x) + |r|^{\eta(x)} + |\xi|^{\delta(x)} , \end{equation} thus, as in \cite{sanchon}, \begin{equation} |f(x,r,\xi)|^{p'(x)} \leq 2^{2 ( p'^{+} - 1)} \Big(g(x)^{p'(x)} + |r|^{p'(x) \eta(x)} + |\xi|^{p'(x) \delta(x)} \Big). \end{equation} Let $E$ be a measurable subset of $\Omega$. Then \begin{equation*} \int_{E} |f(x,v,\nabla v)|^{p'(x)} dx \leq C \Big( \int_{E} g(x)^{p'(x)} dx + \int_{E} |v|^{p'(x) \eta(x)} dx + \int_{E} |\nabla v|^{p'(x) \delta(x)} dx \Big), \end{equation*} with $0 \leq \eta(x) < p(x) - 1$ implying $ 0 \leq p'(x) \eta(x) < p(x)$ and \begin{equation} 0 \leq \delta(x) < \frac{p(x) - 1}{p'(x)} \Rightarrow 0 \leq p'(x) \delta(x) 1,\\ p'^{+} & \text{if } |a(x,u, \nabla v)|_{(L^{p'(x)}(\Omega))^N} \leq 1,\end{cases} $$ we recall that $\|\psi\|$ its equivalent to the norm $|\nabla\psi|_{p(x)}$ on $W_0^{1,p(x)}(\Omega)$ (see Remark\eqref{rem2.2}). We have, $k\in L^{p'(x)}(\Omega)$, $u\in W_0^{1,p(x)}(\Omega)$ and $v\in W_0^{1,p(x)}(\Omega)$. Therefore, \begin{equation} \sum_{i=1}^N \int_{\Omega} a_{i}(x, u, \nabla v) \frac{\partial \psi}{\partial x_{i}} \leq C\|\psi\|. \end{equation} Similarly, \begin{align*} \int_{\Omega} f(x, u, \nabla u) \psi dx &\leq \big( \frac{1}{p^{-}} + \frac{1}{p'^{-}} \big) |f(x, u, \nabla u)|_{L^{p'(x)}(\Omega)} |\psi|_{L^{p(x)}(\Omega)} \\ &\leq \big( \frac{1}{p^{-}} + \frac{1}{p'^{-}} \big) \big( \int_{\Omega} |f(x,u,\nabla u)|^{p'(x)} dx \Big)^{1/\alpha} \|\psi\|, \end{align*} where $$ \alpha = \begin{cases} p'^{-} & \text{if } |f(x, u, \nabla u)|_{L^{p'(x)}(\Omega)} > 1, \\ p'^{+} & \text{if } |f(x, u, \nabla u)|_{L^{p'(x)}(\Omega)} \leq 1. \end{cases} $$ Then, by (H4), \begin{align*} &\int_{\Omega} f(x, u, \nabla u) \psi dx\\ &\leq \big(\frac{1}{p^{-}} + \frac{1}{p'^{-}} \big) \|\psi\| \big[ \int_{\Omega} (g(x) + |u|^{\eta(x)} + |\nabla u|^{\delta(x)} )^{p'(x)} dx \big]^{1/\alpha} \\ &\leq \big(\frac{1}{p^{-}} + \frac{1}{p'^{-}} \big) \|\psi\| 2^{2 (p'^{+} - 1)\frac{1}{\alpha}} \big[ \int_{\Omega} (g(x)^{p'(x)} + |u|^{\eta(x) p'(x)} + |\nabla u|^{\delta(x) p'(x)} ) dx \big]^{1/\alpha} \\ &\leq \big(\frac{1}{p^{-}} + \frac{1}{p'^{-}} \big) \|\psi\| 2^{\frac{2 (p'^{+} - 1)}{\alpha}} \big[ \int_{\Omega} g(x)^{p'(x)} dx + \int_{\Omega} |u|^{\eta(x) p'(x)} dx\\ &\quad + \int_{\Omega} |\nabla u|^{\delta(x) p'(x)} ) dx \big]^{1/\alpha} \\ &\leq \big(\frac{1}{p^{-}} + \frac{1}{p'^{-}} \big) \|\psi\| 2^{\frac{2 (p'^{+} - 1)}{\alpha}} \big[ \int_{\Omega} g(x)^{p'(x)} dx + |u|^{\beta}_{L^{p' \eta }} + |\nabla u|^{\theta}_{L^{p' \delta }} \big]^{1/\alpha}, \end{align*} where $$ \beta = \begin{cases} (\eta p' )^{+} & \text{if } |u|_{L^{p' \eta}} > 1\\ (\eta p' )^{-} & \text{if } |u|_{L^{p' \eta}} \leq 1\,, \end{cases} \quad \theta = \begin{cases} (\delta p' )^{+} & \text{if } |\nabla u|_{L^{p'\delta}} > 1\\ (\delta p' )^{-} & \text{if } |\nabla u|_{L^{p' \delta}} \leq 1\,. \end{cases} $$ Since $ 0\leq \eta (x) < p(x) - 1 $, this implies $ 0\leq \eta(x)p'(x) < p(x)$. Then there exists a constant $C_1 > 0$ such that \begin{equation} |u|_{L^{p' \eta}} \leq C_1 |u|_{L^{p(x)}(\Omega)} \end{equation} and $ 0\leq \delta(x)< (p(x)-1)/p'(x)$, this implies $0\leq \delta(x)p'(x)0$ such that \begin{equation} |\nabla u|_{L^{p' \delta}} \leq C_2 |\nabla u|_{L^{p(x)}(\Omega)}\,. \end{equation} Since $u\in W_0^{1,p(x)}(\Omega)$, there exists a constant $C_3> 0$ such that \begin{equation} \int_{\Omega} f(x, u, \nabla u) \psi dx \leq C_3 \|\psi\|\,. \end{equation} Therefore, there exists a constant $C_0 > 0 $ such that \begin{equation} |\langle B(u,v), \psi \rangle| \leq C_0 \|\psi\| \quad \text{for all } u,v\in W_0^{1,p(x)}(\Omega); \end{equation} i.e., $\langle B(u,v),\psi\rangle$ is bounded in $W_0^{1,p(x)}(\Omega) \times W_0^{1,p(x)}(\Omega)$. We claim that $v \mapsto B(u,v) $ is hemicontinuous for all $u\in W_0^{1,p(x)}(\Omega)$; i.e., the operator $\lambda \mapsto \langle B(u,v_1 + \lambda v_2), \psi \rangle$ is continuous for all $v_1, v_2, \psi \in W_0^{1,p(x)}(\Omega)$. For this, we need Lemma \ref{lem2}. Since $ a_{i} $ is a carath\'eodory function, \begin{equation} a_{i}(x,u, \nabla(v_1 + \lambda v_2)) \to a_{i}(x,u,\nabla v_1) \quad \text{a.e. in $\Omega$ as } \lambda\mapsto 0. \end{equation} and, by (H1), \begin{equation} |a(x,u, \nabla(v_1 + \lambda v_2))| \leq \beta ( k(x) + |u|^{p(x) - 1} + |\nabla (v_1 + \lambda v_2)|^{p(x)- 1})\,. \end{equation} Further, $(a(x,u, \nabla(v_1 + \lambda v_2)))_{\lambda}$ is bounded in $(L^{p'(x)}(\Omega))^N$; thus, by Lemma \ref{lem2}, \begin{equation}\label{eqq.6} a(x,u, \nabla (v_1 + \lambda v_2) ) \rightharpoonup a(x,u, \nabla v_1) \quad \text{in } (L^{p'(x)}(\Omega))^N \text{ as } \lambda\to 0, \end{equation} Hence, \begin{align*} &\lim_{\lambda \to 0} \langle B(u,v_1 + \lambda v_2) , \psi \rangle\\ &= \lim_{\lambda \to 0} \sum_{i=1}^N \int_{\Omega} a_{i}(x,u, \nabla(v_1 + \lambda v_2) ) \frac{\partial \psi}{\partial x_{i}}dx - \int_{\Omega} f(x,u, \nabla u) \psi dx \\ &= \sum_{i=1}^N \int_{\Omega} a_{i}(x,u, \nabla v_1) \frac{\partial \psi}{\partial x_{i}} dx - \int_{\Omega} f(x,u, \nabla u) \psi dx\\ &= \langle B(u,v_1), \psi\rangle \text{ for all } v_1, v_2, \psi\in W_0^{1,p(x)}(\Omega) \end{align*} Similarly, we show that $u \mapsto B(u,v)$ is bounded and hemicontinuous for all $v \in W_0^{1,p(x)}(\Omega)$. Indeed. By (H4), we have $(f(x,u_1 + \lambda u_2, \nabla (u_1 + \lambda u_2)))_{\lambda}$ is bounded in $L^{p'(x)}(\Omega)$, and since $f$ is a carath\'eodory function, \begin{equation} f(x,u_1 + \lambda u_2, \nabla (u_1 + \lambda u_2)) \to f(x,u_1, \nabla u_1) \quad \text{as } \lambda \to 0, \end{equation} Hence, Lemma \ref{lem2} gives \begin{equation}\label{eqq.7} f(x,u_1 + \lambda u_2, \nabla (u_1 + \lambda u_2)) \rightharpoonup f(x,u_1,\nabla u_1) \quad \text{in } L^{p'(x)}(\Omega) \text{ as }\lambda \to 0, \end{equation} On the other hand, as in\eqref{eqq.6}, we have \begin{equation}\label{eqq.8} a(x,u_1 + \lambda u_2, \nabla v) \rightharpoonup a(x,u_1,\nabla v) \quad \text{ in } L^{p'(x)}(\Omega) \text{ as } \lambda\to 0. \end{equation} Combining \eqref{eqq.7} and \eqref{eqq.8}, we conclude that $u \mapsto B(u,v)$ is bounded and hemicontinuous. \noindent\textbf{Assertion 3.} From (H2), we have \begin{equation} \langle B(u,u) - B(u,v), u-v\rangle = \sum_{i = 1}^N \int_{\Omega} ( a_{i}(x,u,\nabla u) - a_{i}(x,u,\nabla v) )\big( \frac{\partial u}{\partial x_{i}} - \frac{\partial v}{\partial x_{i}} \big) dx >0 \end{equation} \noindent\textbf{Assertion 4.} Assume that $u_{n} \rightharpoonup u$ in $W_0^{1,p(x)}(\Omega)$, and $\langle B(u_{n},u_{n})-B(u_{n},u),u_{n}-u\rangle \to 0 \text{ as } n \to \infty$, we claim that $ B(u_{n}, v) \rightharpoonup B(u,v)$ in $W^{-1,p'(x)}(\Omega)$. We have $\langle B(u_{n},u_{n}) - B(u_{n},u), u_{n} - u \rangle \to 0$ as $n\to \infty$, \begin{align*} &\langle \sum_{i = 1}^N - \big[\frac{\partial}{\partial x_{i}} a_{i}(x, u_{n}, \nabla u_{n}) + a_{i}(x, u_{n}, \nabla u) \big], u_{n} - u \rangle \\ &= \sum_{i = 1}^N \int_{\Omega} \big[a_{i}(x, u_{n}, \nabla u_{n}) - a_{i}(x, u_{n}, \nabla u) \big] \big(\frac{\partial u_{n}}{\partial x_{i}} - \frac{\partial u}{\partial x_{i}}\big) dx \to 0 \quad \text{as } n \to \infty \end{align*} Then by Lemma \ref{lem3}, we have $ u_{n} \to u$ in $W_0^{1,p(x)}(\Omega) $ and it follows from Lemma \ref{lem4} that \begin{equation} f(x,u_{n}, \nabla u_{n}) \to f(x,u, \nabla u) \quad \text{in } L^{p'(x)}(\Omega). \end{equation} since $ u_{n} \rightharpoonup u $ in $ W_0^{1,p(x)}(\Omega) $ and $ v\in W_0^{1,p(x)}(\Omega)$, by Lemma \ref{lem1}, $ a_{i}(x,u_{n}, \nabla v) \to a_{i}(x,u,\nabla v)$ in $L^{p'(x)}(\Omega)$. Consequently, \begin{equation} \int_{\Omega} a_{i}(x,u_{n}, \nabla v) \frac{\partial \psi}{\partial x_{i}} dx \to \int_{\Omega} a_{i}(x,u, \nabla v) \frac{\partial \psi}{\partial x_{i}} dx. \end{equation} On the other hand, we have $ f(x,u_{n}, \nabla u_{n}) \to f(x,u, \nabla u) $ in $ L^{p'(x)}(\Omega) $, thus weakly. Since $ \psi \in W_0^{1,p(x)}(\Omega)$, we have $ \psi \in L^{p(x)}(\Omega)$. Then $$ \int_{\Omega} f(x,u_{n}, \nabla u_{n}) \psi dx \to \int_{\Omega} f(x,u, \nabla u) \psi dx \quad \text{as } n \to \infty $$ Therefore, \begin{align*} \lim_{n\to \infty} \langle B(u_{n},v), \psi \rangle &= \lim_{n\to \infty} \Big( \sum_{i=1}^N \int_{\Omega} a_{i}(x,u_{n}, \nabla v ) \frac{\partial \psi}{\partial x_{i}} dx - \int_{\Omega} f(x,u_{n}, \nabla v_{n} ) \psi dx\Big) \\ &= \sum_{i=1}^N \int_{\Omega} a_{i}(x,u, \nabla v ) \frac{\partial \psi}{\partial x_{i}} dx - \int_{\Omega} f(x,u, \nabla u) \psi dx \\ &= \langle B(u,v) , \psi\rangle \quad \text{for all } \psi \in W_0^{1,p(x)}(\Omega). \end{align*} \noindent\textbf{Assertion 5.} Assume $ u_{n} \rightharpoonup u $ in $ W_0^{1,p(x)}(\Omega) $ and $ B(u_{n},v) \rightharpoonup \psi$ in $ W^{-1,p'(x)}(\Omega) $. We claim that $ \langle B(u_{n},v), u_{n}\rangle \to \langle \psi , u\rangle$. Thanks to $ u_{n} \rightharpoonup u $ in $ W_0^{1,p(x)}(\Omega)$, we obtain by Lemma \ref{lem1}, \begin{equation}\label{eq.a5.1} a_{i}(x,u_{n}, \nabla v) \to a_{i}(x,u, \nabla v) \quad \text{in } L^{p'(x)}(\Omega) \text{ as } n \to \infty. \end{equation} Such that \begin{equation}\label{eq.a5.2} \int_{\Omega} a_{i}(x,u_{n}, \nabla v) \frac{\partial u_{n}}{\partial x_{i}} dx \to \int_{\Omega} a_{i}(x,u, \nabla v) \frac{\partial u}{\partial x_{i}} dx. \end{equation} Hence together with \begin{equation}\label{eq.a5.3} \sum_{i=1}^N \int_{\Omega} a_{i}(x,u_{n}, \nabla v ) \frac{\partial u}{\partial x_{i}} dx - \int_{\Omega} f(x,u_{n}, \nabla u_{n}) u dx \to \langle \psi,u\rangle, \end{equation} we have \begin{align*} \langle B(u_{n},v) , u_{n}\rangle &= \sum_{i=1}^N \int_{\Omega} a_{i}(x,u_{n}, \nabla v) \frac{\partial u_{n}}{\partial x_{i}} dx - \int_{\Omega} f(x,u_{n}, \nabla u_{n}) u_{n} dx \\ &= \sum_{i=1}^N \big[\int_{\Omega} a_{i}(x,u_{n}, \nabla v) \big(\frac{\partial u_{n}}{\partial x_{i}} - \frac{\partial u}{\partial x_{i}}\big) dx + \int_{\Omega} a_{i}(x,u_{n}, \nabla v) \frac{\partial u}{\partial x_{i}} dx\big] \\ &\quad - \int_{\Omega} f(x,u_{n}, \nabla u_{n}) u dx -\int_{\Omega} f(x,u_{n}, \nabla u_{n}) (u_{n} - u) dx. \end{align*} But in view of \eqref{eq.a5.1} and \eqref{eq.a5.2}, we obtain \begin{equation}\label{eq.a5.4} \sum_{i=1}^N \int_{\Omega} a_{i}(x,u_{n}, \nabla v) \big(\frac{\partial u_{n}}{\partial x_{i}} -\frac{\partial u}{\partial x_{i}} \big) dx \to 0. \end{equation} On the other hand, by H\"{o}lder's inequality, \begin{align*} &\int_{\Omega} |f(x,u_{n}, \nabla u_{n})(u_{n} - u)| dx \\ &\leq \big( \frac{1}{p^{-}} + \frac{1}{p'^{-}} \big) |f(x,u_{n}, \nabla u_{n})|_{L^{p'(x)}(\Omega)} |u_{n} - u|_{L^{p(x)}(\Omega)} \\ &\leq C |u_{n} - u|_{L^{p(x)}(\Omega)} \to 0 \quad \text{ as } n \to \infty; \end{align*} i.e., \begin{equation}\label{eq.a5.5} \int_{\Omega} f(x,u_{n}, \nabla u_{n})(u_{n} - u) dx \to 0 \quad \text{as } n \to \infty. \end{equation} Thanks to \eqref{eq.a5.3}, \eqref{eq.a5.4} and \eqref{eq.a5.5}, we conclude that \begin{equation} \lim_{n \to \infty} \langle B(u_{n},v) , u_{n}\rangle = \langle \psi , u \rangle. \end{equation} \smallskip \noindent \textbf{Step 2} We claim that the operator $B$ satisfies the coercivity condition \begin{equation} \lim_{\|v\|\to \infty} \frac{\langle B(v), v\rangle }{\|v\|} = +\infty. \end{equation} Since \begin{equation} \langle B(v),v\rangle = \sum_{i = 1}^N \int_{\Omega} a_{i}(x,v,\nabla v)\ \frac{\partial v}{\partial x_{i}} \,dx - \int_{\Omega} f(x,v, \nabla v) v\,dx, \end{equation} Then, by (H3), \begin{equation}\label{s2.1} \langle B v,v\rangle \geq \alpha \|v\|^{p(x)} - \int_{\Omega} f(x,v, \nabla v) v\,dx \end{equation} In view of (H4), \begin{equation} \int_{\Omega} f(x,v,\nabla v)v dx \leq \int_{\Omega} g(x) |v|\,dx + \int_{\Omega} |v|^{\eta(x) + 1} dx + \int_{\Omega} |\nabla v|^{\delta(x)} |v| dx \end{equation} Thanks to H\"{o}lder's inequality, we have \begin{equation}\label{s2.2} \int_{\Omega} g(x) |v| dx \leq \big( \frac{1}{p^{-}} + \frac{1}{p'^{-}}\big) |g|_{L^{p'(x)}(\Omega)} |v|_{L^{p(x)}(\Omega)} \leq C_0\|v\| \end{equation} on the other hand, $$ \int_{\Omega} |v|^{\eta(x) + 1} dx \leq \begin{cases} |v|^{\eta^{+} + 1}_{L^{\eta(x) + 1}(\Omega)} & \text{if } |v|_{L^{\eta(x) + 1}(\Omega)} > 1, \\ |v|^{\eta^{-} + 1}_{L^{\eta(x) + 1}(\Omega)} & \text{if } |v|_{L^{\eta(x) + 1}(\Omega)} \leq 1, \end{cases} $$ Thus, \begin{equation}\label{s2.3} \int_{\Omega} |v|^{\eta(x) + 1} \leq |v|^{\beta}_{L^{\eta(x) + 1} (\Omega)}, \end{equation} where $$ \beta = \begin{cases} \eta^{+} + 1 & \text{if } |v|_{L^{\eta(x) + 1}(\Omega)} > 1, \\ \eta^{-} + 1 & \text{if } |v|_{L^{\eta(x) + 1}(\Omega)} \leq 1, \end{cases} $$ since $ 0 \leq \eta(x) < p(x) - 1 $ implies $ 1 \leq \eta(x) + 1 < p(x) $, consequently, \eqref{s2.3} becomes \begin{equation}\label{s2.4} \int_{\Omega} |v|^{\eta(x) + 1} dx \leq C_1 |v|^{\beta}_{L^{p(x)}(\Omega)}\\ \leq C_1 \|v\|^{\beta}\quad \text{with } \beta < p^{-}. \end{equation} Further, by H\"{o}lder's inequality, \begin{align*} \int_{\Omega} |\nabla v|^{\delta(x)} |v| dx &\leq \big(\frac{1}{p^{-}} + \frac{1}{p'^{-}}\big) \big\|\nabla v|^{\delta(x)}\big|_{L^{p'(x)}(\Omega)} |v|_{L^{p(x)}(\Omega)} \\ &\leq \big(\frac{1}{p^{-}} + \frac{1}{p'^{-}}\big) \Big(\int_{\Omega} |\nabla v|^{\delta(x) p'(x)} dx \Big)^{1/\gamma} |v|_{L^{p(x)}(\Omega)} \\ &\leq \big(\frac{1}{p^{-}} + \frac{1}{p'^{-}}\big) \Big(\int_{\Omega} |\nabla v|^{\theta} dx\Big)^{1/\gamma} |v|_{L^{p(x)}(\Omega)}, \end{align*} where $$ \gamma = \begin{cases} p'^{-} & \text{if } \big\|\nabla v|^{\delta(x)} \big|_{L^{p'(x)}(\Omega)} > 1, \\ p'^{+} & \text{if } \big\|\nabla v|^{\delta(x)} \big|_{L^{p'(x)}(\Omega)} \leq 1, \end{cases} \quad \theta = \begin{cases} \delta^{+} p'^{+} & \text{if } |\nabla v| > 1, \\ \delta^{-} p'^{-} & \text{if } |\nabla v| \leq 1. \end{cases} $$ Then \begin{equation} \int_{\Omega} |\nabla v|^{\delta(x)} |v| dx \leq C \big(| v|_{W_0^{1,\theta}(\Omega)}\big)^{\theta/\gamma} |v|_{L^{p(x)}(\Omega)}, \end{equation} since $0 \leq \delta (x) < (p(x) - 1) / p'(x)$ implies $0 \leq \delta(x) p'(x) < p(x) - 1 $, and $$ 0\leq \delta^{+} < \big(\frac{p-1}{p'}\big)^{-} = \frac{p^{-} - 1}{p'^{+}} \Longrightarrow 0 \leq \delta^{+} p'^{+} < p^{-} - 1, $$ and $$ 0 \leq \delta^{-} p'^{-} < \frac{(p^{-} - 1)}{p'^{+}} p'^{-} \leq p^{-} - 1. $$ Therefore, $0 \leq \theta < p^{-} -1 < p(x)$. On the other hand, $$ 0 \leq \frac{\theta}{p'^{+}} < \frac{p^{-} - 1}{p'^{+}} \quad \text{and} \quad 0 \leq \frac{\theta}{p'^{-}} < \frac{p^{-} - 1}{p'^{-}}\,. $$ Thus \begin{equation}\label{s2.5} \int_{\Omega} |\nabla v|^{\delta(x)} |v| dx \leq C_2 \|v\|^{\theta/\gamma} \|v\| \end{equation} Combining \eqref{s2.1}, \eqref{s2.2}, \eqref{s2.4}, and \eqref{s2.5}, we deduce that \begin{equation} \frac{\langle B(v),v \rangle}{\|v\|} \geq \alpha \|v\|^{p(x) - 1} - C_0 - C_1 \|v\|^{\beta - 1} - C_2 \|v\|^{\theta/\gamma}\,. \end{equation} Then we have $$ 0 \leq \frac{\theta}{p'^{+}} < \frac{p^{-} - 1}{p'^{+}}, \quad 0 \leq \frac{\theta}{p'^{-}} < \frac{p^{-} - 1}{p'^{-}}, \quad \frac{p^{-} - 1}{p'^{+}} \leq \frac{p^{-} - 1}{p'^{-}}\,; $$ Thus, \begin{equation} 0 \leq \frac{\theta}{\gamma} < \frac{p^{-} - 1}{p'^{-}} < p^{-} - 1\,. \end{equation} Since $\beta - 1 < p ^{-}- 1$, we conclude that $$ \frac{\langle B(v),v\rangle}{\|v\|} \geq \alpha \|v\|^{p(x) - 1} - C_0 - C_1 \|v\|^{\beta - 1} - C_2 \|v\|^{\theta/\gamma } \to + \infty\quad \text{as } \|v\| \to + \infty. $$ Finally, by a classical theorem in \cite{lions}, the problem $\eqref{pb1}$ has a solution, so the proof of theorem \ref{thm4.1} is achieved. \begin{thebibliography}{00} \bibitem{rhoudaf} Y. 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