\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2009(2009), No. 72, pp. 1--12.\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{9mm}} \begin{document} \title[\hfilneg EJDE-2009/72\hfil Existence of solutions] {Existence of solutions for a $p(x)$-Laplacian non-homogeneous equations} \author[I. Andrei \hfil EJDE-2009/72\hfilneg] {Ionic\u{a} Andrei} \address{Ionic\u{a} Andrei \newline Department of Mathematics, High School of Cujmir, 227150 Cujmir, Romania} \email{andreiionica2003@yahoo.com} \thanks{Submitted March 9, 2009. Published June 2, 2009.} \subjclass[2000]{35D05, 35J60, 58E05} \keywords{$p(x)$-Laplace operator; generalized Lebesgue-Sobolev space; \hfill\break\indent critical point; weak solution} \begin{abstract} We study the boundary value problem \begin{gather*} -\mathop{\rm div}(|\nabla u|^{p(x)-2}\nabla u)=f(x,u)\quad \text{in }\Omega, \\ u=0\quad \text{on }\partial \Omega, \end{gather*} where $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$. Our attention is focused on the cases when $$f(x,u)=\pm (-\lambda |u|^{p(x)-2}u+|u|^{q(x)-2}u),$$ where $p(x)1\}.$ For any $h\in C_+(\overline\Omega)$ we define $$h^+=\sup_{x\in\Omega}h(x)\quad\mbox{and}\quad h^-=\inf_{x\in\Omega}h(x).$$ For $p\in C_+(\overline\Omega)$, we introduce {\it the variable exponent Lebesgue space} \begin{align*} L^{p(\cdot)}(\Omega)=\big\{&u: u \mbox{ is a measurable real-valued function}\\ &\text{such that }\int_\Omega|u(x)|^{p(x)}\,dx<\infty\big\}, \end{align*} endowed with the so-called {\it Luxemburg norm} $$|u|_{p(\cdot)}=\inf\big\{\mu>0;\;\int_\Omega| \frac{u(x)}{\mu}|^{p(x)}\,dx\leq 1\big\},$$ which is a separable and reflexive Banach space. For basic properties of the variable exponent Lebesgue spaces we refer to \cite{ko}. If $0 <|\Omega|<\infty$ and $p_1$, $p_2$ are variable exponents in $C_+(\overline\Omega)$ such that $p_1 \leq p_2$ in $\Omega$, then the embedding $L^{p_2(\cdot)}(\Omega)\hookrightarrow L^{p_1(\cdot)}(\Omega)$ is continuous, \cite[Theorem~2.8]{ko}. Let $L^{p'(\cdot)}(\Omega)$ be the conjugate space of $L^{p(\cdot)}(\Omega)$, obtained by conjugating the exponent pointwise that is, $1/p(x)+1/p'(x)=1$, \cite[Corollary~2.7]{ko}. For any $u\in L^{p(\cdot)}(\Omega)$ and $v\in L^{p'(\cdot)}(\Omega)$ the following H\"older type inequality $$\label{Hol} \big|\int_\Omega uv\,dx\big|\leq\Big(\frac{1}{p^-}+ \frac{1}{{p'}^-}\Big)|u|_{p(\cdot)}|v|_{p'(\cdot)}$$ is valid. An important role in manipulating the generalized Lebesgue-Sobolev spaces is played by the {\it $p(\cdot)$-modular} of the $L^{p(\cdot)}(\Omega)$ space, which is the mapping $\rho_{p(\cdot)}:L^{p(\cdot)}(\Omega)\to\mathbb{R}$ defined by $$\rho_{p(\cdot)}(u)=\int_\Omega|u|^{p(x)}\,dx.$$ If $(u_n)$, $u\in L^{p(\cdot)}(\Omega)$ then the following relations hold \begin{gather}\label{L40} |u|_{p(\cdot)}<1\;(=1;\,>1)\;\Leftrightarrow\;\rho_{p(\cdot)}(u) <1\;(=1;\,>1) \\ \label{L4} |u|_{p(\cdot)}>1 \;\Rightarrow\; |u|_{p(\cdot)}^{p^-}\leq\rho_{p(\cdot)}(u) \leq|u|_{p(\cdot)}^{p^+} \\ \label{L5} |u|_{p(\cdot)}<1 \;\Rightarrow\; |u|_{p(\cdot)}^{p^+}\leq \rho_{p(\cdot)}(u)\leq|u|_{p(\cdot)}^{p^-} \\ \label{L6} |u_n-u|_{p(\cdot)}\to 0\;\Leftrightarrow\;\rho_{p(\cdot)} (u_n-u)\to 0, \end{gather} since $p^+<\infty$. For a proof of these facts see \cite{ko}. Spaces with $p^{+}=\infty$ have been studied by Edmunds, Lang and Nekvinda \cite{ed}. Next, we define $W_0^{1,p(x)}(\Omega)$ as the closure of $C_0^{\infty}(\Omega)$ under the norm $\| u\|_{p(x)}=|\nabla u|_{p(x)}.$ The space $(W_0^{1,p(x)}(\Omega),\| \cdot \|_{p(x)})$ is a separable and reflexive Banach space. We note that if $q\in C_+(\overline{\Omega})$ and $q(x)0$. First, we consider the problem $$\label{e6} \begin{gathered} -\mathop{\rm div}(|\nabla u|^{p(x)-2}\nabla u) =-\lambda | u | ^{p(x)-2}u +| u | ^{q(x)-2}u \quad\text{in } \Omega \\ u=0 \quad\text{on }\partial\Omega \end{gathered}$$ We say that $u\in W_0^{1,p(x)}(\Omega)$ is a weak solution of \eqref{e6} if $\int_{\Omega}| \nabla u | ^{p(x)-2}\nabla u \nabla v dx +\lambda \int_{\Omega} | u | ^{p(x)-2}uv\,dx- \int_{\Omega}| u | ^{q(x)-2}uv\,dx=0$ for all $v\in W_0^{1,p(x)}(\Omega)$. We will prove the following result. \begin{theorem}\label{thm1} For every $\lambda >0$, problem \eqref{e6} has infinitely many weak solutions provided $2\leq p^-$, $p^+0$ such that for any $\lambda \geq \lambda^*$ problem \eqref{e7} has a nontrivial weak solution provided $p^+0)$ is coercive in $H_0^1(\Omega)$, Ambrosetti and Rabinowitz showed that problem \eqref{e8} has a positive solution for any $\lambda >0$. \section{Proof of Theorem 1} The key argument in the proof is the following version of the Mountain Pass Theorem (see \cite[Theorem 9.12]{ra}): \subsection*{Mountain Pass Theorem} Let $X$ be an infinite dimensional real Banach space and let $I\in C^1(X,\mathbb{R})$ be even, satisfying the Palais-Smale condition (i.e., any sequence $\{x_n\}\subset X$ such that $\{I(x_n)\}$ is bounded and $I'(x_n)\to 0$ in $X^*$ has a convergent subsequence) and $I(0)=0$. Suppose that \begin{itemize} \item[(I1)] there exists two constants $\rho$, $a>0$ such that $I(x)\geq a$ if $\|x\|=\rho$, \item[(I2)] for each finite dimensional subspace $X_1\subset X$, the set $\{x\in X_1; I(x)\geq 0\}$ is bounded. \end{itemize} Then $I$ has an unbounded sequence of critical values. \smallskip Let $E$ denote the generalized Sobolev space $W_0^{1,p(x)}(\Omega)$ and let $\lambda >0$ be arbitrary but fixed. The energy functional corresponding to problem \eqref{e6} is defined as $J_{\lambda}:E\to \mathbb{R}$, $J_{\lambda}(u)=\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}dx +\lambda \int_{\Omega}\frac{1}{p(x)}|u|^{p(x)}dx -\int_{\Omega}\frac{1}{q(x)}|u|^{q(x)}dx.$ A simple calculation based on relations \eqref{L4} and \eqref{L5} and the compact embedding of $E$ into $L^{r(x)}(\Omega)$ for all $r\in C_+(\overline{\Omega})$ with $r(x)0$ and $\alpha >0$ such that $J_{\lambda}(u)\geq \alpha >0$ for any $u\in E$ with $\|u\|_{p(x)}=\eta$ \end{lemma} \begin{proof} We first point out that we have $$\label{e11} |u(x)|^{q^-}+|u(x)|^{q^+}\geq |u(x)|^{q(x)}, \quad \forall x\in \overline{\Omega}$$ Using \eqref{e11} we deduce that $$\label{e12} J_{\lambda}(u)\geq \frac{1}{p^+}\cdot\int_{\Omega}|\nabla u|^{p(x)}dx-\frac{1}{q^-}\cdot \Big(\int_{\Omega}|u|^{q^-}dx+\int_{\Omega}|u|^{q^+}dx\Big)$$ Since $p^+0$ there exists a positive constant $K_2(\lambda)$ such that $$\label{e19} \lambda \cdot \int_{\Omega}\frac{1}{p(x)}|u|^{p(x)}dx\leq K_2(\lambda)\cdot\left(\|u\|_{p(x)}^{p^-}+\|u\|_{p(x)}^{p^+}\right), \quad \forall u\in E.$$ By inequalities \eqref{e15} and \eqref{e19}, we get $J_{\lambda}(u)\leq K_1 \cdot \left( \|u\|_{p(x)}^{p-}+\|u\|_{p(x)}^{p^+}\right)+K_2(\lambda)\cdot \left(\|u\|_{p(x)}^{p^-}+\|u\|_{p(x)}^{p^+}\right) -\frac{1}{q^+}\int_{\Omega}|u|^{q(x)}dx$ for all $u\in E$. Let $u\in E$ be arbitrary but fixed. We define $\Omega_1=\{x\in \Omega; |u(x)|<1\}, \quad \Omega_2 =\Omega\setminus \Omega_1.$ Then we have \begin{align*} J_{\lambda}(u) &\leq K_1 \cdot \left( \|u\|_{p(x)}^{p-}+\|u\|_{p(x)}^{p^+}\right)+K_2(\lambda)\cdot \left(\|u\|_{p(x)}^{p^-}+\|u\|_{p(x)}^{p^+}\right) -\frac{1}{q^+}\int_{\Omega}|u|^{q(x)}dx\\ & \leq K_1 \cdot \left( \|u\|_{p(x)}^{p-}+\|u\|_{p(x)}^{p^+}\right)+K_2(\lambda)\cdot \left(\|u\|_{p(x)}^{p^-}+\|u\|_{p(x)}^{p^+}\right) -\frac{1}{q^+}\int_{\Omega_2}|u|^{q(x)}dx\\ &\leq K_1 \cdot \left( \|u\|_{p(x)}^{p-}+\|u\|_{p(x)}^{p^+}\right) +K_2(\lambda)\cdot \left(\|u\|_{p(x)}^{p^-}+\|u\|_{p(x)}^{p^+}\right) -\frac{1}{q^+}\int_{\Omega_2}|u|^{q^-}dx\\ &\leq K_1 \cdot \left( \|u\|_{p(x)}^{p-}+\|u\|_{p(x)}^{p^+}\right) +K_2(\lambda)\cdot \left(\|u\|_{p(x)}^{p^-}+\|u\|_{p(x)}^{p^+}\right)\\ &\quad -\frac{1}{q^+}\int_{\Omega}|u|^{q^-}dx+\frac{1}{q^+} \int_{\Omega_1}|u|^{q^-}dx. \end{align*} But there exists a positive constant $K_3$ such that $\frac{1}{q^+}\int_{\Omega_1}|u|^{q^-}dx\leq K_3,\quad \forall u\in E.$ Thus we deduce that $J_{\lambda}(u)\leq K_1 \cdot \big( \|u\|_{p(x)}^{p-}+\|u\|_{p(x)}^{p^+}\big) +K_2(\lambda)\cdot \big(\|u\|_{p(x)}^{p^-}+\|u\|_{p(x)}^{p^+}\big) -\frac{1}{q^+}\int_{\Omega}|u|^{q^-}dx+K_3,$ for all $u\in E$. The functional $|\cdot|_{q^-}:E\to \mathbb{R}$ defined by $|u|_{q^-}=\Big( \int_{\Omega}|u|^{q^-}dx\Big)^{1/q^-}$ is a norm in $E$. In the finite dimensional subspace $E_1$ the norms $|\cdot|_{q^-}$ and $\|\cdot\|_{p(x)}$ are equivalent, so there exists a positive constant $K=K(E_1)$ such that $\|u\|_{p(x)}\leq K\cdot |u|_{q^-}, \quad \forall u\in E_1.$ As a consequence we have that there exists a positive constant $K_4$ such that $J_{\lambda}(u)\leq K_1 \cdot \left( \|u\|_{p(x)}^{p-} +\|u\|_{p(x)}^{p^+}\right)+K_2(\lambda)\cdot \left(\|u\|_{p(x)}^{p^-}+\|u\|_{p(x)}^{p^+}\right) -K_4\cdot \|u\|^{q^-}+K_3,$ for all $u\in E_1$. Hence $K_1 \cdot \left(\|u\|_{p(x)}^{p-}+\|u\|_{p(x)}^{p^+}\right) +K_2(\lambda)\cdot \left(\|u\|_{p(x)}^{p^-}+\|u\|_{p(x)}^{p^+}\right) -K_4\cdot \|u\|^{q^-}_{p(x)}+K_3\geq 0,$ for all $u\in S$. and since $q^->p^+$ we conclude that $S$ is bounded in $E$. The proof is complete. \end{proof} \begin{lemma}\label{lem3} If $\{u_n\}\subset E$ is a sequence which satisfies the conditions \begin{gather}\label{e20} |J_{\lambda}(u_n)|1$for any integer$n$. By \eqref{e21} we deduce that there exists$N_1>0$such that for any$n>N_1$, we have $\|J_{\lambda}'(u_n)\|\leq 1.$ On the other hand, for any$n>N_1$fixed, the application $E\ni v\to \langle J'_{\lambda}(u_n),v \rangle$ is linear and continuous. The above information implies $|\langle J'_{\lambda}(u_n), v \rangle| \leq \|J_{\lambda}'(u_n)\|\cdot \|v\|_{p(x)}\leq \|v\|_{p(x)}, \quad \forall v\in E, n>N_1.$ Setting$v=u_n$we have $-\|u_n\|_{p(x)}\leq \int_{\Omega}|\nabla u_n|^{p(x)}dx +\lambda \int_{\Omega}|u_n|^{p(x)}dx-\int_{\Omega}|u_n|^{q(x)}dx \leq \|u_n\|_{p(x)}$ for all$n>N_1$. We obtain $$\label{e22} -\|u_n\|_{p(x)}- \int_{\Omega}|\nabla u_n|^{p(x)}dx-\lambda \int_{\Omega}|u_n|^{p(x)}dx\leq -\int_{\Omega}|u_n|^{q(x)}dx$$ for any$n>N_1$. Provided that$\|u_n\|_{p(x)}>1relations \eqref{e20}, \eqref{e22} and \eqref{L4} imply \begin{align*} M&>J_{\lambda}(u_n)\\ &\geq \Big( \frac{1}{p^+}-\frac{1}{q^-}\Big)\cdot \int_{\Omega}(|\nabla u_n|^{p(x)})dx\\ &\quad +\lambda \cdot \Big( \frac{1}{p^+}-\frac{1}{q^-}\Big)\cdot \int_{\Omega}|u_n|^{p(x)}dx-\frac{1}{q^-}\cdot \|u_n\|_{p(x)}\\ &\geq \Big( \frac{1}{p^+}-\frac{1}{q^-}\Big)\cdot \int_{\Omega} |\nabla u_n|^{p(x)}dx-\frac{1}{q^-}\cdot \|u_n\|_{p(x)}\\ &\geq \Big( \frac{1}{p^+}-\frac{1}{q^-}\Big) \cdot \|u_n\|^{p^-}_{p(x)}-\frac{1}{q^-}\cdot \|u_n\|_{p(x)}. \end{align*} Lettingn\to \infty$we obtain a contradiction. It follows that$\{u_n\}$is bounded in$E$. Since$\{u_n\}$is bounded in$E$we deduce that there exists a subsequence, again denoted by$\{u_n\}$, and$u_0\in E$such that$\{u_n\}$converges weakly to$u_0$in$E$. Using Theorem 1.3 in \cite{fa},$E$is compactly embedded in$L^{p(x)}(\Omega)$and in$L^{q(x)}(\Omega)$it follows that$\{u_n\}$converges strongly to$u_0$in$L^{p(x)}(\Omega)$and$L^{q(x)}(\Omega). The above information and relation \eqref{e21} imply $\langle J'_{\lambda}(u_n)-J'_{\lambda}(u_0), u_n-u_0\rangle \to 0 \quad \text{as } n\to \infty.$ On the other hand, we have \begin{align*} &\int_{\Omega}(|\nabla u_n|^{p(x)-2}\nabla u_n-|\nabla u_0|^{p(x)-2} \nabla u_0)\cdot (\nabla u_n-\nabla u_0)dx\\ &=\langle J'_{\lambda}(u_n)-J'_{\lambda}(u_0), u_n-u_0 \rangle -\lambda \cdot \int_{\Omega}(|u_n|^{p(x)-2}u_n-|u_0|^{p(x)-2}u_0) (u_n-u_0)dx\\ &\quad +\int_{\Omega}(|u_n|^{q(x)-2}u_n-|u_0|^{q(x)-2}u_0)(u_n-u_0)dx. \end{align*} Using the fact that\{u_n\}$converges strongly to$u_0$in$L^{q(x)}(\Omega)and inequality \eqref{Hol}, we have \begin{align*} &\big|\int_{\Omega}(|u_n|^{q(x)-2}u_n-|u_0|^{q(x)-2}u_0)(u_n-u_0)dx\big|\\ &\leq\big|\int_{\Omega}|u_n|^{q(x)-2}u_n(u_n-u_0)dx\big| +\big|\int_{\Omega}|u_0|^{q(x)-2}u_0(u_n-u_0)dx\big|\\ &\leq C_3\cdot \|u_n|^{q(x)-1}|_{\frac{q(x)}{q(x)-1}}\cdot |u_n-u_0|_{q(x)} +C_4\cdot \|u_0|^{q(x)-1}|_{\frac{q(x)}{q(x)-1}}\cdot |u_n-u_0|_{q(x)} \end{align*} whereC_3$and$C_4$are positive constants. Since$|u_n-u_0|_{q(x)}\to 0$as$n\to \infty$we deduce that \begin{gather}\label{e24} \lim_{n\to \infty}\int_{\Omega}(|u_n|^{q(x)-2}u_n-|u_0|^{q(x)-2}u_0) (u_n-u_0)dx=0,\\ \label{e25} \lim_{n\to \infty}\int_{\Omega}(|u_n|^{p(x)-2}u_n-|u_0|^{p(x)-2}u_0) (u_n-u_0)dx=0. \end{gather} By \eqref{e24} and \eqref{e25}, we obtain $$\label{e26} \lim_{n\to \infty}\int_{\Omega}(|\nabla u_n|^{p(x)-2}\nabla u_n -|\nabla u_0|^{p(x)-2}\nabla u_0)\cdot (\nabla u_n -\nabla u_0)dx=0.$$ It is known that $$\label{e27} (|z|^{r-2}z-|t|^{r-2}t)\cdot (z-t)\geq \big(\frac{1}{2}\big)^r|z-t|^r, \quad \forall r\geq 2,\; z,t\in \mathbb{R}^N.$$ Relations \eqref{e26} and \eqref{e27} yield $\lim_{n\to \infty}\int_{\Omega}|\nabla u_n-\nabla u_0|^{p(x)}dx=0\,.$ This fact and relation \eqref{L6} imply$\|u_n-u_0\|_{p(x)}\to \infty$as$n\to \infty$. The proof is complete. \end{proof} \begin{proof}[Completed proof of Theorem \ref{thm1}] It is clear that the functional$J_{\lambda}$is even and verifies$J_{\lambda}(0)=0$. Lemma \ref{lem3} implies that$J_{\lambda}$satisfies the Palais-Smale condition. On the other hand, Lemmas \ref{lem1} and \ref{lem2} show that conditions (I1) and (I2) are satisfied. The Mountain Pass Theorem can be applied to the functional$J_{\lambda}$. We conclude that equation \eqref{e6} has infinitely many weak solutions in$E$. The proof is complete. \end{proof} \section{Proof of Theorem \ref{thm2}} Let$E$denote the generalized Sobolev space$W_0^{1,p(x)}(\Omega)$and let$\lambda >0$be arbitrary but fixed. We start by introducing the energy functional corresponding to problem \eqref{e6} as$I_{\lambda}:E\to \mathbb{R}$, $I_{\lambda}(u)=\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}dx -\lambda \int_{\Omega}\frac{1}{p(x)}|u|^{p(x)}dx +\int_{\Omega}\frac{1}{q(x)}|u|^{q(x)}dx.$ The same arguments as those used in the case of the functional$J_{\lambda}$show that$I_{\lambda}$is well-defined on$E$and$I_{\lambda}\in C^1(E,\mathbb{R})$with the derivative given by $\langle I'_{\lambda}(u), v\rangle =\int_{\Omega}|\nabla u|^{p(x)-2}\nabla u \nabla v dx -\lambda \int_{\Omega}|u|^{p(x)-2}uv\,dx +\int_{\Omega}|u|^{q(x)-2}uv dx$ for any$u$,$v\in E$. We obtain that the weak solutions of \eqref{e6} are the critical points of$I_{\lambda}$. This time our idea is to show that$I_{\lambda}$possesses a nontrivial global minimum point in$E$. With this end in view we start by proving two auxiliary results. \begin{lemma}\label{lem4} The functional$I_{\lambda}$is coercive on$E$. \end{lemma} \begin{proof} To prove this lemma, we first show that for any$a$,$b>0$and$00it follows that $a-b\cdot t^{l-k}<0, \quad \forall t>\big(\frac{a}{b}\big)^{1/(l-k)}$ and t^k \cdot(a-b\cdot t^{l-k})\leq a\cdot t^k1, \begin{align*} I_{\lambda}(u) &\geq \frac{1}{p^+}\int_{\Omega}|\nabla u|^{p(x)}dx -\frac{\lambda}{p^-}\int_{\Omega}|u|^{p(x)}dx +\frac{1}{q^+}\int_{\Omega}|u|^{q(x)}dx\\ &\geq \frac{1}{p^+}\|u\|^{p^-}_{p(x)} -\Big( \frac{\lambda}{p^-}\int_{\Omega}|u|^{p(x)}dx -\frac{1}{q^+}\int_{\Omega}|u|^{q(x)}dx\Big)\\ &\geq \frac{1}{p^+}\|u\|_{p(x)}^{p^-}-D. \end{align*} Thus I_{\lambda} is coercive and the proof of is complete. \end{proof} \begin{lemma}\label{lem5} The functional I_{\lambda} is weakly lower semicontinuous. \end{lemma} \begin{proof} First we prove that the functional A:E\to \mathbb{R}, \[ A(u)=\int_{\Omega}\frac{1}{p^(x)}|\nabla u|^{p(x)}dx, is convex. Indeed, since the function[0,\infty)\ni t\to t^s$is convex for any$s>1$, we deduce that for each$x\in \Omega$fixed it the inequality $\big|\frac{z+t}{2}\big|^{p(x)}\leq \big| \frac{|z|+|t|}{2}\big|^{p(x)} \leq \frac{1}{2}|z|^{p(x)}+\frac{1}{2}|t|^{p(x)},\quad \forall z,\; t \in \mathbb{R}^N$ holds. Using the above inequality we deduce that $\big| \frac{\nabla u +\nabla v}{2}\big|^{p(x)} \leq \frac{1}{2}|\nabla u|^{p(x)}+\frac{1}{2}|\nabla v|^{p(x)}, \quad \forall u,v\in E, \; x\in \Omega .$ Multiplying with$1/p(x)$and integrating over$\Omega$we obtain $A\big(\frac{u+v}{2}\big)\leq \frac{1}{2}A(u)+\frac{1}{2}A(v), \quad \forall u,v\in E.$ Thus$A$are convex. Next, we show that the functional$A$is weakly lower semicontinuous on$E$. Taking into account that$A$is convex, by \cite[Corollary III.8]{br} it is sufficient to show that$A$is strongly lower semicontinuous on$E$. We fix$u\in E$and$\varepsilon>0$. Let$v\in E$be arbitrary. Since$Ais convex and inequality \eqref{Hol} holds; we have \begin{align*} A(u)&\geq A(u)+\langle A'(u),v-u\rangle\\ &\geq A(u)-\int_{\Omega}|\nabla u|^{p(x)-1}|\nabla (v-u)|dx\\ &\geq A(u)-D_1\cdot \|\nabla u|^{p(x)-1}|_{\frac{p(x)}{p(x)-1}} \cdot |\nabla (u-v)|_{p(x)}\\ &\geq A(u)-D_2\cdot \|u-v\|_{p(x)}\\ &\geq A(u)-\varepsilon \end{align*} for allv\in E$with$\|u-v\|_{p(x)}<\varepsilon /[\|\nabla u|^{p(x)-1}|_{\frac{p(x)}{p(x)-1}}]$. We have denoted by$D_1$and$D_2$two positive constants. It follows that$A$is strongly lower semicontinuous and since it is convex we obtain that$A$is weakly lower semicontinuous. Finally, we remark that if$\{u_n\}\subset E$is a sequence which converges weakly to$u$in$E$then$\{u_n\}$converges strongly to$u$in$L^{p(x)}(\Omega)$and$L^{q(x)}(\Omega)$. Thus,$I_{\lambda}$is weakly lower semicontinuous. The proof is complete. \end{proof} \begin{proof}[Proof of Theorem \ref{thm2}] By Lemmas \ref{lem4} and \ref{lem5}, we deduce that$I_{\lambda}$is coercive and weakly lower semicontinuous on$E$. Then \cite[Theorem 1.2]{st} implies that there exist a global minimizer$u_{\lambda}\in E$of$I_{\lambda}$and thus a weak solution of problem \eqref{e7}. We show that$u_{\lambda}$is not trivial for$\lambda$large enough. Indeed, letting$t_0>1$be a fixed real and$\Omega_1$an open subset of$\Omega$with$|\Omega_1|>0$we deduce that there exists$u_0\in C^{\infty}_0(\Omega)\subset E$such that$u_0(x)=t_0$for any$x\in \overline{\Omega}_1$and$0\leq u_0(x)\leq t_0$in$\Omega \setminus \Omega_1. We have \begin{align*} I_{\lambda}(u_0) &=\int_{\Omega}\frac{1}{p(x)}|\nabla u_0|^{p(x)}dx -\lambda \int_{\Omega}\frac{1}{p(x)}|u_0|^{p(x)}dx +\int_{\Omega}\frac{1}{q(x)}|u_0|^{q(x)}dx\\ &\leq L-\frac{\lambda}{p^+}\int_{\Omega_1}|u_0|^{p(x)}dx\\ &\leq L-\frac{\lambda}{p^+}\cdot t_0^{p^-}\cdot |\Omega_1| \end{align*} whereL$is a positive constant. Thus, there exists$\lambda^*>0$such that$I_{\lambda}(u_0)<0$for any$\lambda \in [\lambda^*,\infty)$. It follows that$I_{\lambda}(u_{\lambda})<0$for any$\lambda\geq \lambda^*$and thus$u_{\lambda}$is a nontrivial weak solution of problem \eqref{e7} for$\lambda$large enough. The proof of is complete. \end{proof} \subsection*{Remark} After this article was accepted, the author learned that the results here are a particular case of the results in \cite{mii}. \begin{thebibliography}{99} \bibitem{ac} E. Acerbi and G. Mingione: Regularity results for a class of functionals with nonstandard growth. Arch. Rational Mech. 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