\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2011 (2011), No. 90, pp. 1--15.\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-2011/90\hfil Compactness results] {Compactness results for quasilinear problems with variable exponent on the whole space} \author[O. Allegue, A. K. Souayah\hfil EJDE-2011/90\hfilneg] {Olfa Allegue, Asma Karoui Souayah} % in alphabetical order \address{Olfa Allegue \newline Institut Pr\'eparatoire aux Etudes d'ing\'enieurs de Tunis\\ 2 Rue Jawaher Lel Nehru, 1008, Montfleury-Tunis, Tunisia} \email{allegue\_olfa@yahoo.fr} \address{Asma Karoui Souayah\newline Institut Pr\'eparatoire aux Etudes d'ing\'enieurs de Tunis\\ 2 Rue Jawaher Lel Nehru, 1008, Montfleury-Tunis, Tunisia} \email{asma.souayah@yahoo.fr} \thanks{Submitted May 2, 2011. Published July 6, 2011.} \subjclass[2000]{35J655, 35J60, 35J70, 58E05} \keywords{$p(x)$-Laplace operator; critical Sobolev exponent; compactness; \hfill\break\indent Palais-Smale condition} \begin{abstract} In this work we give a compactness result which allows us to prove the point-wise convergence of the gradients of a sequence of solutions to a quasilinear inequality and for an arbitrary open set. This result suggests solutions to many problems, notably nonlinear elliptic problems with critical exponent. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{proposition}[theorem]{Proposition} \allowdisplaybreaks \section{Introduction and preliminary results} In their recent work El Hamidi and Rakotoson \cite{hami} gave a compactness result to prove the point-wise convergence of the gradients of a sequence of solutions to a general quasilinear inequality and for an arbitrary open set. They proved the following result. \begin{lemma}\label{leml} Let $\hat{a}$ be a Carath\'eodory function from $ \mathbb{R}^N\times \mathbb{R} \times \mathbb{R}$ into $\mathbb{R}^N$ satisfying the usual Leray-Lions growth and monotonicity conditions. Let $(u_n)$ be a bounded sequence of $W^{1,p}_{\rm loc}(\mathbb{R}^N)=\{v \in L^{p}_{\rm loc}(\mathbb{R}^N), |\nabla v | \in L^{p}_{\rm loc}(\mathbb{R}^N) \}$, with $11 \text{ for all }x\in\overline\Omega\}. $$ For any $h\in C_+(\overline{\Omega})$ we define $$ h^+=\sup_{x\in\Omega}h(x)\quad\text{and}\quad h^-=\inf_{x\in\Omega}h(x). $$ For any $p\in C_+(\overline{\Omega})$, we define the variable exponent Lebesgue space \[ L^{p(x)}(\Omega)=\{u: \text{$u$ is a Borel real-valued function on } \Omega, \int_\Omega|u(x)|^{p(x)}\,dx<\infty \}. \] We define on $L^{p(x)}$, the so-called \emph{Luxemburg norm}, by the formula $$ |u|_{p(x)}:=\inf\big\{\mu>0:\int_\Omega|\frac{u(x)}{\mu}|^{p(x)}\,dx \leq 1\big\}. $$ Variable exponent Lebesgue spaces resemble classical Lebesgue spaces in many aspects: they are separable and Banach spaces \cite[Theorem 2.5; Corollary 2.7]{KR} and the H\"older inequality holds \cite[Theorem 2.1]{KR}. The inclusions between Lebesgue spaces are also naturally generalized \cite[Theorem 2.8]{KR}: if $0 <|\Omega|<\infty$ and $r_1$, $r_2$ are variable exponents so that $r_1(x) \leq r_2(x)$ almost everywhere in $\Omega$ then there exists the continuous embedding $L^{r_2(x)}(\Omega)\hookrightarrow L^{r_1(x)}(\Omega)$. We denote by $L^{p'(x)}(\Omega)$ the conjugate space of $L^{p(x)}(\Omega)$, where $1/p(x)+1/p'(x)=1$. For any $u\in L^{p(x)}(\Omega)$ and $v\in L^{p'(x)}(\Omega)$ the H\"older type inequality \begin{equation}\label{eq1} \big|\int_\Omega uv\,dx\big|\leq\big(\frac{1}{p^-}+ \frac{1}{{p'}^-}\big)|u|_{p(x)}|v|_{p'(x)}, \end{equation} is held. An important role in manipulating the generalized Lebesgue-Sobolev spaces is played by the \emph{modular} of the $L^{p(x)}(\Omega)$ space, which is the mapping $\rho_{p(x)}:L^{p(x)}(\Omega)\to \mathbb{R}$ defined by $$ \rho_{p(x)}(u)=\int_\Omega|u|^{p(x)}\,dx. $$ The space $W^{1,p(x)}(\Omega)$ is equipped by the norm $$ \|u\|=| u|_{p(x)}+|\nabla u|_{p(x)}. $$ We recall that if $(u_n), u, \in W^{1,p(x)}(\Omega)$ and $p^+<\infty$ then the following relations hold: \begin{gather}\label{eq2} \min(|u|_{p(x)}^{p^{-}},|u|_{p(x)}^{p^{+}})\leq \rho_{p(x)}(u)\leq \max(|u|_{p(x)}^{p^{-}},|u|_{p(x)}^{p^{+}}),\\ \label{eq3} \min(|\nabla u|_{p(x)}^{p^{-}},|\nabla u|_{p(x)}^{p^{+}})\leq \rho_{p(x)}(|\nabla u|)\leq \max(|\nabla u|_{p(x)}^{p^{-}}, |\nabla u|_{p(x)}^{p^{+}}),\\ \label{eq4} \begin{gathered} |u|_{p(x)}\to 0\;\Leftrightarrow\;\rho_{p(x)} (u)\to 0,\\ \lim_{n\to\infty}|u_n-u|_{p(x)}=0\; \Leftrightarrow\;\lim_{n\to\infty}\rho_{p(x)}(u_n-u)=0,\\ |u_n|_{p(x)}\to \infty\;\Leftrightarrow\;\rho_{p(x)} (u_n)\to \infty. \end{gathered} \end{gather} We define also $W_{0}^{1,p(x)}(\Omega)$ as the closure of $C_{0}^{\infty}(\Omega)$ under the norm $$ \|u\|_{p(x)}=|\nabla u|_{p(x)}. $$ The space $(W_{0}^{1,p(x)}(\Omega),\|\cdot\|)$ is a separable and reflexive Banach space. Next, we recall some embedding results regarding variable exponent Lebesgue-Sobolev spaces. We note that if $s(x)\in C_+(\overline{\Omega})$ and $s(x)< p^*(x)$ for all $x\in \overline{\Omega} $ then the embedding $W_0^{1,p(x)}(\Omega)\hookrightarrow L^{s(x)}(\Omega)$ is compact and continuous, where $p^*(x)=Np(x)/(N-p(x))$ if $p(x) 0$, $\sigma\in \mathbb{R}$, let \[ S_{\epsilon}(\sigma)=\begin{cases} \sigma &\text{if } |\sigma|\leq\epsilon\\ \epsilon \operatorname{sign}(\sigma) &\text{otherwise}, \end{cases} \] and $\sigma^k:=S_k(\sigma)$ for $k\geq 1$. We shall consider a nonlinear map $\hat{a}:\Omega\times \mathbb{R}\times \mathbb{R}^N\to\mathbb{R}^N$ satisfying the following conditions: \begin{itemize} \item[(L1)] $\hat{a}(x,.,.)$ is a continuous map for almost every $x$ and for all $(\sigma,\xi)\in\mathbb{R}\times \mathbb{R}^N$, $\hat{a}(.,\sigma,\xi)$ is measurable (such a property is called Carath\'eodory property), \item[(L2)] $\hat{a}$ maps bounded sets of $W^{1,p(x)}_{\rm loc}(\Omega)$ into bounded sets of $L^{p'(x)}_{\rm loc}(\Omega)$, and for almost all $x\in\Omega$, for all $(\sigma, \xi)$ in $\mathbb{R}\times\mathbb{R}^N$, $\hat{a}(x,\sigma,\xi)\cdot\xi\geq0$, for almost every $x\in\Omega$ and for all $v\in W^{1,p(x)}_{\rm loc}(\Omega)$, the mapping $u \mapsto\hat{a}(x,u,\nabla v)$ is continuous from $W^{1,p(x)}(\omega)$-weak into $L^{p'(x)}(\omega)$-strong, for all $\omega\subset\subset\Omega$, \item[(L3)] for almost every $x\in\Omega$, for all $(\sigma,\xi_i)\in\mathbb{R}\times\mathbb{R}^N$, $i=1,2$, $$ [\hat{a}(x,\sigma,\xi_1)-\hat{a}(x,\sigma,\xi_2)][\xi_1-\xi_2]>0, \quad \text{for } \xi_1\neq\xi_2. $$ \item[(L4)] if for some $x\in\Omega$, there is a sequence $(\sigma_n,\xi_{1n})\in\mathbb{R}\times\mathbb{R}^N$, $\xi_2\in\mathbb{R}^N$ such that $[\hat{a}(x,\sigma_n,\xi_{1n})-\hat{a}(x,\sigma_n,\xi_2)] [\xi_{1n}-\xi_2]$ and $\sigma_n$ are bounded as $n\to +\infty$ then $|\xi_{1n}|$ remains in a bounded set of $\mathbb{R}$ as $n\to+\infty$. \end{itemize} As a corollary of our main result, we state the following result. \begin{lemma}\label{lem1} Let $\hat{a}$ be a Carath\'eodory function from $ \mathbb{R}^N\times \mathbb{R} \times \mathbb{R}$ into $\mathbb{R}^N$ satisfying the usual Leray-Lions growth and monotonicity conditions. Let $(u_n)$ be a bounded sequence of $$ W^{1,p(x)}_{\rm loc}(\mathbb{R}^N) =\{v \in L^{p(x)}_{\rm loc}(\mathbb{R}^N), |\nabla v | \in L^{p(x)}_{\rm loc}(\mathbb{R}^N) \}, $$ with $10$: $$ \limsup_{n\to +\infty}\int_{\Omega} \hat{a}(x,u_n(x), \nabla u_n(x))\cdot \nabla(\phi S_{\epsilon}(u_n-u^k)) \leq o(1) $$ as $\epsilon\to 0$ then there exists a subsequence still denoted $(u_n)$ such that $$ \nabla u_n(x) \to \nabla u(x)\quad \text{ a.e. in } \Omega \text{ as } n \to +\infty. $$ \end{itemize} \end{theorem} \begin{remark} \label{rmk1}\rm \begin{enumerate} \item The term $o(1)$ in (ii) might depend on $k$ and $\phi$. \item (L2) is satisfied if for all $\omega\subset\subset\Omega$, there is a constant $c_{\omega}>0$ and a function $a_0\in L^{p'(x)}(\omega)$ such that for almost every $x\in\omega$, for all $(\sigma,\xi)\in\mathbb{R}\times\mathbb{R}^N$: $$ |\hat{a}(x,\sigma,\xi)|\leq c_{\omega}[|\sigma|^{p(x)-1} +|\xi|^{p(x)-1}+a_0(x)]. $$ and (L4) is true if $\hat{a}(x,\sigma,\xi)\cdot\xi \geq c_{\omega}^1|\xi|^{p(x)}-c_{\omega}^2$, $c_{\omega}^1>0$. \item Bounded sets in $W_{\rm loc}^{1,p(x)}(\Omega)$ will be bounded in $$ W^{1,p(x)}(\omega)=\{v\in L^{p(x)}(\omega), \nabla v\in L^{p(x)}(\omega)\},\quad\text{for every } \omega \subset\subset \Omega. $$ \end{enumerate} \end{remark} \begin{proof}[Proof of theorem \ref{thm1}] (i) Let $(w_j)_{j\geq 0}$ be a sequence of bounded relatively compact subsets of $\Omega$ such that $\overline{\omega}_j\subset \omega_{j+1}$ and $ \cup_{j=0}^{+\infty}\omega_j=\Omega$. Since $(u_n)_n$ is bounded in $W^{1,p(x)}(\omega_j)$, by the usual embeddings, we deduce that there is a subsequence $u_{n_{j}}$ and a function $u$ in $W^{1,p(x)}(\omega_j)$ such that $u_{n_{j}}(x)\to u(x)$ as $n\to \infty$. We conclude with the usual diagonal Cantor process. (ii) Let $\phi\in C_c^{\infty}(\Omega), 0\leq \phi\leq 1, \phi=1$ on $\omega_j$ and supp$(\phi)\subset\omega_{j+1}$, and set $$ \Delta(u_n,u)(x)=[\hat{a}(x,u_n(x),\nabla u_n(x)) -\hat{a}(x,u_n(x),\nabla u(x))]\nabla (u_n-u)(x). $$ Then one has: \begin{itemize} \item[(ii.1)] $\Delta(u_n,u)(x)\geq 0$ a.e. on $\Omega$ (due to (L3)). \item[(ii.2)] $ \sup_n \int_{\omega_{j+1}} \Delta (u_n,u)dx$ is finite (since $(u_n)$ is in a bounded set of $W^{1,p(x)}_{\rm loc}(\Omega)$ and the growth condition (L2)). \end{itemize} Let us show that $ \lim_n\int_{\Omega}\phi\Delta (u_n,u)^{\frac{1}{p(x)}}dx=0$. On one hand, \begin{equation}\label{R1} \int_{\Omega}\phi\Delta (u_n,u)^{\frac{1}{p(x)}}dx =\int_{\{|u|>k\}}\phi\Delta (u_n,u)^{\frac{1}{p(x)}}dx +\int_{\{|u|\leq k\}}\phi\Delta (u_n,u)^{\frac{1}{p(x)}}dx. \end{equation} By the H\"older inequality \[ \int_{\{|u|>k\}}\Delta (u_n,u)^{\frac{1}{p(x)}}\phi\,dx \leq |\Delta(u_n,u)^{\frac{1}{p(x)}}|_{p(x)} |\phi|_{\frac{p(x)}{p(x)-1}} \leq a_1(j)|\phi|_{\frac{p(x)}{p(x)-1}}, \] where $a_s(j)$ are different constants depending on $j$ but independent of $n$, $\epsilon$ and $k$. Noticing that $$ \operatorname{meas}\{x\in w_{j+1}:|u|>k\}\leq\frac{c_1(j)}{k^{p^-}}, $$ one deduces that \begin{equation}\label{R2} \rho_{\frac{p(x)}{p(x)-1}}(\phi) =\int_{\{|u|>k\}}\phi^{\frac{p(x)}{p(x)-1}}dx \leq \frac{c_1(j)}{k^{p^-}} \end{equation} where $c_m(j)$ are different constants depending on $j$ and $\phi$ but independent of $n$, $\epsilon$ and $k$. We conclude that \begin{equation}\label{asm} \limsup_{n\to \infty}\int_{\{|u|>k\}}\Delta (u_n,u)^{\frac{1}{p(x)}} \phi\,dx\leq o(1)\quad \text{as } k\to \infty \end{equation} While for the second integral, we have \begin{equation}\label{R3} \begin{split} \int_{\{|u|\leq k\}}\Delta (u_n,u)^{\frac{1}{p(x)}}\phi\,dx &=\int_{\{|u|\leq k\}\cap \{|u_n-u|\leq \epsilon|\}} \Delta (u_n,u)^{\frac{1}{p(x)}}\phi\,dx\\ &\quad +\int_{\{|u|\leq k\} \cap \{|u_n-u|>\epsilon\}}\Delta (u_n,u)^{\frac{1}{p(x)}}\phi\,dx. \end{split} \end{equation} Moreover, the second term in the right hand side in the last inequality satisfies \[ \int_{\{|u|\leq k\}\cap \{|u_n-u|>\epsilon\}} \Delta (u_n,u)^{\frac{1}{p(x)}}\phi\,dx \leq |\Delta(u_n,u)^{\frac{1}{p(x)}}|_{p(x)} |\phi|_{\frac{p(x)}{p(x)-1}}\\ \leq a_2(j) |\phi|_{\frac{p(x)}{p(x)-1}} \] and $$ \rho_{\frac{p(x)}{p(x)-1}}(\phi)\leq a_2(\phi) \operatorname{meas} \{x\in w_{j+1}: |u_n-u|>\epsilon\}. $$ Since $(u_n)$ converges to $u$ in measure, we deduce that, for $n$ sufficiently large, $\operatorname{meas} \{x\in w_{j+1}: |u_n-u|>\epsilon\}\leq \epsilon$. It follows that \begin{equation}\label{R4} \limsup_{n\to +\infty}\int_{\{|u|\leq k\}\cap \{|u_n-u|>\epsilon\}}\Delta (u_n,u)^{\frac{1}{p(x)}}\phi\,dx \leq o(1) \quad \text{as }\epsilon\to 0. \end{equation} Setting $A_{n,k}^{\epsilon} =w_{j+1}\cap\{|u|\leq k\}\cap \{|u_n-u|\leq\epsilon\}$, we obtain from the H\"older inequality \begin{equation} \label{R5} \int_{A_{n,k}^{\epsilon}}\Delta (u_n,u)^{\frac{1}{p(x)}}\phi\,dx \leq c_2(j) |\Delta(u_n,u)^{\frac{1}{p(x)}}\phi^{\frac{1}{p(x)}} |_{p(x)}, \end{equation} and \[ \rho_{p(x)}(\Delta(u_n,u)^{\frac{1}{p(x)}}\phi^{\frac{1}{p(x)}}) =I_{n,k}^1(\epsilon)-I_{n,k}^2(\epsilon), \] with \begin{gather*} I_{n,k}^1(\epsilon)=\int_{A_{n,k}^{\epsilon}} \hat{a}(x,u_n,\nabla u_n)\cdot\nabla(u_n-u)\phi\,dx,\\ I_{n,k}^2(\epsilon)=\int_{\{|u|\leq k\}} \hat{a}(x,u_n,\nabla u) \cdot\nabla S_{\epsilon}(u_n-u)\phi\,dx. \end{gather*} Since $\hat{a}(x,u_n,\nabla u)\to \hat{a}(x,u,\nabla u)$ strongly in $L^{p'(x)}(w_{j+1})$ (by the last statement of (L2)) and $\nabla S_{\epsilon}(u_n-u)\rightharpoonup 0$ in $L^{p(x)}(w_{j+1})$-weak, we deduce that \begin{equation}\label{R6} \lim_{n\to +\infty} I_{n,k}^2(\epsilon)=0, \end{equation} while for the term $I_{n,k}^1(\epsilon)$, we obtain \begin{equation} I_{n,k}^1(\epsilon)\leq\int_{\Omega}\hat{a}(x,u_n,\nabla u_n) \cdot\nabla(\phi S_{\epsilon}(u_n-u^k))-\int_{\Omega} \hat{a}(x,u_n,\nabla u_n)\cdot\nabla\phi S_{\epsilon}(u_n-u^k)dx. \end{equation} Since \begin{equation} \big|\int_{\Omega}\hat{a}(x,u_n,\nabla u_n)\cdot\nabla \phi S_{\epsilon}(u_n-u^k)dx\big|\leq c_3(j)\epsilon; \end{equation} then assumption (ii) implies \begin{equation}\label{R9} \limsup_{n\to +\infty} I_{n,k}^1(\epsilon) \leq c_3(j)\epsilon+\circ (1) \quad \text{as } \epsilon\to 0. \end{equation} Combining relations \eqref{R5}, \eqref{R6} and \eqref{R9}, it follows that \begin{equation}\label{R10} \limsup_{n\to +\infty}\int_{A_{n,k}^{\epsilon}} \Delta (u_n,u)^{\frac{1}{p(x)}}\phi\,dx\leq o(1)\quad \text{as }\epsilon\to 0. \end{equation} Letting first $\epsilon\to 0$ and then $k$ to infinity, by relations \eqref{R1}, \eqref{asm}, \eqref{R3}, \eqref{R4} and \eqref{R10}, we deduce $$ \lim_{n\to +\infty}\int_{\Omega}\Delta (u_n,u)^{\frac{1}{p(x)}} \phi\,dx=0. $$ We then obtain that for a subsequence $(u_{j_n})$, $$ \Delta (u_{j_n},u)(x)\to 0 \quad \text{a.e. on } w_j. $$ Arguing as Leray-Lions \cite{JLL, jll}, we deduce from (L4) that $\nabla u_{j_n}(x)\to \nabla u(x)$ a.e. on $w_j$. The proof is achieved by the diagonal process of Cantor. \end{proof} \begin{proof}[Proof of lemma \ref{lem1}] Since $(u_n)$ belongs to a bounded set of $W_{\rm loc}^{1,p(x)}(\mathbb{R}^N)$, statement (i) of Theorem \ref{thm1} implies that there is a function $u$ and a subsequence still denoted by $(u_n)$ such that $$ u_n(x)\to u(x)\quad \text{a.e. in $\mathbb{R}^N$, as } n\to \infty $$ and $$ u\in W_{\rm loc}^{1,p(x)}(\mathbb{R}^N). $$ Then for all $\phi\in C_c^{\infty}(\mathbb{R}^N)$, $\phi S_{\epsilon}(u_n-u^k)$ is an element of $W_{\rm comp}^{1,p(x)}(\mathbb{R}^N)$ and \begin{equation} \big|\int_{\mathbb{R}^N}f_n\phi S_{\epsilon}(u_n-u^k)dx\big| \leq \epsilon |\phi|_{\infty}|f_n|_{L^1(\omega)}\leq c(\phi)\epsilon, \end{equation} (for every $\phi$ such that $\operatorname{supp}(\phi)\subset\omega$, $\overline{\omega}$ is a compact of $\mathbb{R}^N$), and $$ |\langle g_n,\phi S_{\epsilon}(u_n-u^k)\rangle| \leq|g_n|_{W^{-1,p'(x)}(\omega)}|\phi S_{\epsilon} (u_n-u^k)|_{W^{1,p(x)}(\mathbb{R}^N)}. $$ Using the fact that $|\phi S_{\epsilon}(u_n-u^k)|_{W^{1,p(x)}(\mathbb{R}^N)}$ is bounded independently of $\epsilon$, $n$, $k$ and that $|g_n|_{W^{-1,p'(x)}(\omega)}\to 0$, it holds: $$ \limsup_n\int_{\mathbb{R}^N}\hat{a}(x,u_n,\nabla u_n)\cdot \nabla(\phi S_{\epsilon}(u_n-u^k))dx\leq O(\epsilon). $$ Finally, by Theorem \ref{thm1} we complete the proof. \end{proof} \section{Examples of applications} In this section, we are interested in the existence of solutions to the problem \begin{equation}\label{P1} \begin{gathered} -\mathop{\rm div}\big(\left( |\nabla u(x)|^{p(x)-2}\right) \nabla u(x)\big) = \lambda f(u)+g(u) \quad \text{for }x\in\Omega,\\ u\geq 0 \quad \text{for } x\in\Omega,\\ u=0 \quad \text{for } x\in\partial\Omega, \end{gathered} \end{equation} where $\Omega \subset\mathbb{R}^N$, ($N\geq3$) is a bounded domain with smooth boundary, $\lambda $ is a positive real number and $p$ is a continuous function on $\overline{\Omega}$ with $p^+0$ and $q:\bar{\Omega}\to \mathbb{R} $ a continuous function such that $$ C_1 t^{q(x)-1}\leq f(t) \leq C_2 t^{q(x)-1}, \quad \forall t\geq 0. $$ \item[(G1)] There exists a positive constant $C_3>0$ such that $$ |g(t)|\leq C_3 |t|^{p^{*}(x)-1},\quad \forall t\in \mathbb{R}. $$ \item[(G2)] There exists $\gamma \in (p^{+},p^{*-}]$ such that $$ 0<\gamma G(t)\leq tg(t),\quad \forall t\in \mathbb{R}, $$ where $G(t)=\int^{t}_{0}g(s)ds$. \end{itemize} We prove the following result. \begin{theorem}\label{thm2} If $10$ problem \eqref{P1} has infinitely many weak solutions provided that $p^{*-}>max(p^{+},q^{+})$. \end{theorem} \subsection*{Proof of Theorem \ref{thm2}} Let $E$ denote the generalized Sobolev space $W^{1,p(x)}_{0}(\Omega)$. The energy functional corresponding to \eqref{P1} is $J_{\lambda}: E \to \mathbb{R}$, defined as $$ J_{\lambda}(u):=\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}dx-\lambda\int_{\Omega}F(u_{+})dx- \int_{\Omega}G(u_{+})dx, $$ where $u_{+}(x)=\max\{u(x),0\} $ and $F$ is defined by $F(t)=\int^{t}_{0}f(s)ds$. \begin{remark} \label{rmk2} \rm Assume that condition (G1) is fulfilled, it is clear that for every $t\geq 0$, we obtain $$ -\frac{C_3}{p^{*-}}t^{p^{*}(x)}\leq G(t)\leq \frac{C_3}{p^{*-}}t^{p^{*}(x)} $$ \end{remark} \begin{proposition} \label{prop1} The functional $J_{\lambda}$ is well-defined on $E$ and $J_{\lambda} \in C^{1}(E,\mathbb{R})$. \end{proposition} \begin{proof} We have the following continuous embedding (see \cite[Theorem 2.8]{KR}) $$ W^{1,p(x)}_{0}(\Omega)\hookrightarrow L^{p^{*}(x)}(\Omega) $$ using the fact that $\Omega$ is bounded, we obtain the continuous embedding $$ W^{1,p(x)}_{0}(\Omega)\hookrightarrow L^{s(x)}(\Omega), \quad s\in [1,p^{*}], $$ which implies that $J_{\lambda}$ is well-defined on $E$ and $J_{\lambda}\in C^{1}(E,\mathbb{R})$, with the derivative given by $$ \langle dJ_{\lambda}(u),v\rangle=\int_{\Omega} \big(|\nabla u|^{p(x)-2}\nabla u \nabla v-\lambda f(u) v -g(u) v \big)dx,\quad \forall v \in E. $$ \end{proof} The proof of Theorem \ref{thm2} is related to Ekeland's variational principle. In order to apply it we need the following lemmas: \begin{lemma}\label{lem2} Under hypotheses of theorem \ref{thm2}, there exists $M_2>0$ such that for all $\rho \in(0,1)$ for all $C_3<\frac{q^-}{p^+ M^{p^{*-}}_2}\rho^{p^+-q^-}$, there exists $\lambda^{*}>0$ and $r>0$ such that, for all $u\in E$ with $\|u\|=\rho$, $J_{\lambda}(u)\geq r>0$ for all $\lambda \in(0,\lambda^{*})$. \end{lemma} \begin{proof} Since $E\hookrightarrow L^{q(x)}(\Omega)$ and $E\hookrightarrow L^{p^{*}(x)}(\Omega)$ are continuous, there exists $M_1>0$ and $M_2>0$ such that \begin{equation}\label{eq23} |u|_{{q(x)}}\leq M_1 \|u\| \quad \text{and} \quad |u|_{{p^{*}(x)}}\leq M_2 \|u\|, \quad \forall u\in E. \end{equation} We fix $\rho \in (0,1)$ such that $ \rho<\min (1,1/M_1,1/M_2)$. Then for all $u \in E$, with $\|u\|=\rho $, we deduce that $$ |u|_{{q(x)}}<1 \quad \text{and} \quad |u|_{{p^*(x)}}<1. $$ Furthermore, by \eqref{eq2} for all $u \in E$ with $\|u\|=\rho $, we have $$ \int_{\Omega}|u|^{q(x)}\,dx\leq|u|^{q^-}_{q(x)},\quad \text{and} \quad \int_{\Omega}|u|^{p^*(x)}\,dx\leq|u|^{p^{*-}}_{p(x)}. $$ The above inequality and relation \eqref{eq23} imply that for all $u \in E$ with $\|u\|=\rho $, \begin{equation}\label{eq24} \int_{\Omega}|u|^{q(x)}\,dx\leq M_1^{q^-}\|u\|^{q^-}, \quad and \quad \int_{\Omega}|u|^{p^*(x)}\,dx\leq M_2^{p^{*-}}\|u\|^{p^{*-}}. \end{equation} Using relation \eqref{eq24} we deduce that, for any $u \in E$ with $\|u\|=\rho$, the following inequalities hold: \begin{align*} J_{\lambda}(u) &\geq \frac{1}{p^{+}}\|u\|^{p^{+}}- \frac{\lambda}{q^{-}}C_2M_1^{q^{-}}\|u\|^{q^{-}} -\frac{C_3}{p^{*-}}M_2^{p^{*-}}\|u\|^{p^{*-}},\\ &\geq \frac{1}{p^{+}}\rho^{p^{+}}-\frac{\lambda}{q^{-}}C_2 M_1^{q^{-}}\rho^{q^{-}}-\frac{C_3}{p^{*-}}M_2^{p^{*-}}\rho^{p^{*-}}. \end{align*} By the above inequality we remark that if we define for all $C_3<\frac{q^-}{p^+ M^{p^{*-}}_2}\rho^{p^+-q^-}$ \begin{equation} \label{eq25} \lambda^{*}=\frac{q^{-}}{2C_2M_1^{q^{-}}} \big[ \frac{1}{p^+}\rho^{p^{+}-q^{-}}-\frac{C_3}{q^-} M^{p^{*-}}_2\big], \end{equation} then for any $\lambda \in (0,\lambda^*)$, there exists $ r>0$ such that $J_{\lambda}(u)\geq r>0$ for all $u\in E$ with $\|u\|=\rho$. The proof is complete. \end{proof} \begin{lemma}\label{lem3} There exists $\phi \in E$ such that $\phi \geq0$, $\phi \neq 0$ and $J_{\lambda}(t\phi)<0$, for $t>0$ small enough. \end{lemma} \begin{proof} Since $q^-0$ be such that $q^-+\epsilon_00$. In fact if $$ \int_{\Omega}|\nabla \phi|^{p(x)}dx+C_3\int_{\Omega}|\phi|^{p^*(x)}dx =0, $$ then $\int_{\Omega}|\phi|^{p^*(x)}dx =0$. Using relation \eqref{eq2}, we deduce that $|\phi|_{p^*(x)}=0$ and consequently $\phi=0$ in $\Omega$ which is a contradiction. The proof is complete. \end{proof} \subsection*{Proof of theorem \ref{thm2}} Let $\lambda^*$ be defined as in \eqref{eq25} and $\lambda \in (0,\lambda^{*})$. By Lemma \ref{lem2} it follows that on the boundary of the ball centered at the origin and of radius $\rho$ in $E$, denoted by $B_{\rho}(0)$, we have \begin{equation}\label{eq26} \inf_{\partial B_{\rho}(0)}J_{\lambda}>0. \end{equation} On the other hand, by Lemma \ref{lem3}, there exists $\phi \in E$ such that $J_{\lambda}(t\phi)<0$, for all $t>0$ small enough. Moreover, relations \eqref{eq2} and \eqref{eq23} imply, that for any $u \in B_{\rho}(0)$, we have \[ J_{\lambda}(u)\geq\frac{1}{p^{+}}\|u\|^{p^{+}} -\frac{\lambda}{q^{-}}C^{q^-}_2M^{q^-}_1\|u\|^{q^{-}} -\frac{C^{p^{*-}}_3}{q^{-}}M^{q^-}_2\|u\|^{p^{*-}}. \] It follows that $$ -\infty0$ and any $v \in B_1(0)$. The above relation yields $$ \frac{J_{\lambda}(u_\epsilon+t\cdot v)-J_{\lambda}(u_\epsilon)}{t} + \epsilon\cdot\| v\|\geq 0. $$ Letting $t\to 0$ it follows that $\langle dJ_{\lambda}(u_\epsilon),v\rangle +\epsilon\cdot\| v \| \geq0 $ we have $\| dJ_{\lambda}(u_\epsilon)\| \leq\epsilon $. We deduce that there exists a sequence $\{w_n\}\subset B_{\rho}(0)$ such that \begin{equation}\label{eq27} J_{\lambda}(u_n) \to J_{\infty}\quad \text{and} \quad dJ_{\lambda}(u_n)\to 0_{E^{*}}. \end{equation} From where we can conclude that $\{u_n\}$ is a bounded $(PS)_{J_{\infty}}$ sequence to $J_{\lambda}$. By a subsequence still denoted by $\{u_n\}$, we may assume that $\{u_n\}$ has a weak limit $u_{\lambda}\in W^{1,p(x)}_{0}(\Omega)$. Moreover, from the definition of the functional $J_{\lambda}$, we can assume that $\{u_n\}$ is a sequence of non negative functions. Now, we need the following lemma. \begin{lemma}\label{lem4} The weak limit $u_{\lambda}$ of $\{u_n\}$ is a non negative solution to \eqref{P1} for $\lambda \in (0,\lambda^*)$. \end{lemma} \begin{proof} In what follows, we will show $dJ_{\lambda}(u_{\lambda})=0$ and $u_{\lambda}\neq 0, \forall \lambda \in (0,\lambda^*)$ which imply that lemma \ref{lem4} holds true. Firstly note that \begin{gather*} J_{\lambda}(u_n)=\int_{\Omega}\frac{1}{p(x)}|\nabla u_n|^{p(x)}dx-\lambda\int_{\Omega}F(u_{n+})dx- \int_{\Omega}G(u_{n+})dx, \\ \langle dJ_{\lambda}(u_n),u_n\rangle=\int_{\Omega}|\nabla u_n|^{p(x)}dx-\lambda\int_{\Omega}f(u_n)u_n\,dx - \int_{\Omega}g(u_n) u_n\,dx. \end{gather*} Then $J_{\lambda}(u_n)-\frac{1}{\gamma}\langle dJ_{\lambda} (u_n),u_n\rangle=J_{\infty}+o_n(1)$. Thus, since $uf(u)\geq0$ for every $u\geq0$, we obtain \begin{align*} &\int_{\Omega}\frac{1}{p(x)}|\nabla u_n|^{p(x)}dx-\frac{1}{\gamma}\int_{\Omega}|\nabla u_n|^{p(x)}dx-\lambda\int_{\Omega}F(u_{n+})dx\\ &+ \frac{\lambda}{\gamma}\int_{\Omega}f(u_n)u_n\,dx - \int_{\Omega}G(u_{n+})dx +\frac{1}{\gamma}\int_{\Omega}g(u_n) u_n\,dx\\ &\geq \big(\frac{1}{p^+}-\frac{1}{\gamma}\big) \int_{\Omega}\frac{1}{p(x)}|\nabla u_n|^{p(x)}dx -\lambda\int_{\Omega} F(u_{n+})dx\\ &\quad +\frac{1}{\gamma} \Big(\int_{\Omega}\big(g(u_n) u_n -\gamma G(u_{n+})\big)dx\Big). \end{align*} Since $\gamma>p^+$ and applying (G2) we have \begin{align*} J_{\lambda}(u_n)-\frac{1}{\gamma}\langle dJ_{\lambda}(u_n), u_n\rangle &\geq \big(\frac{1}{p^+}-\frac{1}{\gamma}\big)\int_{\Omega}\frac{1}{p(x)}|\nabla u_n|^{p(x)}dx-\lambda\int_{\Omega}F(u_{n+})dx,\\ &\geq -\lambda\int_{\Omega}F(u_{n+})dx,\\ &\geq -\lambda C_2\int_{\Omega}\frac{1}{q(x)} u_n^{q(x)}dx. \end{align*} Using \eqref{eq2} we deduce that $ -\frac{\lambda C_2}{q^+} | u_n |^{q^+}_{q(x)}\leq J_{\infty}+o_n(1)$. Moreover $W^{1,p(x)}_{0}(\Omega)\hookrightarrow L^{q(x)}(\Omega)$ is compact and passing to the limit as $n \to \infty$, we obtain $$ -\frac{\lambda C_2}{q^+} | u_{\lambda}|^{q^+}_{q(x)}\leq J_{\infty}<0. $$ We deduce that $u_{\lambda}\neq 0$. To conclude that $u_{\lambda}$ is a solution to \eqref{P1}, we use Theorem \ref{thm1}, which implies $\nabla u_n (x)\to \nabla u_\lambda (x)$ a.e. in $\Omega$ as $n\to \infty$. \end{proof} \subsection*{Proof of Theorem \ref{thm3}} Now, we are concerned with the special case of problem \eqref{P1}, \begin{equation}\label{P2} \begin{gathered} -\mathop{\rm div}\big(\big( |\nabla u(x)|^{p(x)-2}\big)\nabla u(x)\big) =-\lambda |u|^{q(x)-2}u+|u|^{p^{*}(x)-2}u \quad \text{for } x\in\Omega,\\ u\geq0 \quad \text{for } x\in\Omega,\\ u=0 \quad \text{for } x\in\partial\Omega, \end{gathered} \end{equation} where $\Omega \subset\mathbb{R}^N$, ($N\geq3$) is a bounded domain with smooth boundary, $\lambda $ is a positive real number and $p$ is a continuous function on $\overline{\Omega}$. The proof of Theorem \ref{thm3} is based on a $\mathbb{Z}_2$-symmetric version for even functionals of the Mountain pass Theorem (see \cite[Theorem 9.12]{Rab}). The energy functional corresponding to the problem \eqref{P2} is $J_{\lambda}:E\to \mathbb{R}$, defined as $$ J_{\lambda}(u):=\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}dx+\lambda\int_{\Omega}\frac{1}{q(x)}|u|^{q(x)}dx - \int_{\Omega}\frac{1}{p^*(x)}|u|^{p^*(x)}dx, $$ It is clear that the functional $J_{\lambda}$ is well-defined on $E$ and $J_{\lambda} \in C^{1}(E,\mathbb{R})$, with the derivative given by $$ \langle dJ_{\lambda}(u),v\rangle =\int_{\Omega}\big(|\nabla u|^{p(x)-2}\nabla u \nabla v +\lambda|u|^{q(x)-2}u v - |u|^{p^*(x)-2}u v \big)dx,\quad \forall v \in E. $$ To use the mountain pass theorem, we need the following lemmas: \begin{lemma}\label{lem5} For any $\lambda> 0$ there exists $r,a >0$ such that $J_{\lambda}(u)\geq a>0$ for any $u \in E$ with $\|u\|=r $. \end{lemma} \begin{proof} Recall that $E$ is continuously embedded in $L^{p^*(x)}(\Omega)$. So there exists a positive constant $C_4$ such that, for all $u\in E$, \begin{equation}\label{eq6} |u|_{{p^*(x)}}\leq C_4 \|u\|. \end{equation} Suppose that $\|u\|<\min(1,\frac{1}{C_4})$, then for all $u \in E$ with $\|u\|=\rho $ we have $|u|_{{p^*(x)}}<1$. Furthermore, relation \eqref{eq2} yields for all $u \in E$ with $\|u\|=\rho $ we have $$ \int_{\Omega}|u|^{p^*(x)}\,dx\leq|u|^{p^{*-}}_{p^*(x)}. $$ The above inequality and relation \eqref{eq6} imply that for all $u \in E$ with $\|u\|=\rho $, we have \begin{equation}\label{eq7} \int_{\Omega}|u|^{p^*(x)}\,dx\leq C_4^{p^{*-}}\|u\|^{p^{*-}}. \end{equation} Then using relation \eqref{eq7}, we deduce that, for any $u \in E$ with $\|u\|=\rho $, the following inequalities hold \begin{align*} J_{\lambda}(u) &\geq \frac{1}{p^{+}}{\int_{\Omega}|\nabla u|^{p(x)}dx}-\frac{1}{p^{*-}}{\int_{\Omega}|u|^{p^*(x)}\,dx},\\ &\geq \frac{1}{p^{+}}\|u\|^{p^{+}}-\frac{1}{p^{*-}}C_4^{p^{*-}} \|u\|^{p^{*-}}. \end{align*} Let $ h(t)=\frac{1}{p^{+}}t^{p^{+}} -\frac{1}{p^{*-}}C_4^{p^{*-}}t^{p^{*-}}$, $t>0$. It is easy to see that $h(t)>0$ for all $t\in(0,t_1)$, where $ t_1<\big(\frac{p^{*-}}{ p^{+} C_4^{p^{*-}}}\big)^{\frac{1}{p^{*-} - p^+}}$.\\ So for any $\lambda >0$, we can choose $r,a>0$ such that $J_{\lambda}(u)\geq a>0$ for all $u\in E$ with $\|u\|=r$. The proof is complete. \end{proof} \begin{lemma}\label{lem6} If $E_1\subset E $ is a finite dimensional subspace, the set $S=\{u \in E_1; J_{\lambda}(u)\geq0\}$ is bounded in $E$. \end{lemma} \begin{proof} We have \begin{equation}\label{eq9} \int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}\,dx\leq K_1(\|u\|^{p^{-}}+\|u\|^{p^{+}} ) \quad \forall u\in E, \end{equation} where $K_1$ is a positive constant. Also we have \begin{equation}\label{eq13} \int_{\Omega}| u|^{q(x)}\,dx\leq |u|^{q^{-}}_{q(x)}+ |u|^{q^{+}}_{q(x)} \quad \forall u\in E. \end{equation} The fact that $E$ is continuously embedded in $L^{q(x)}(\Omega)$, assures the existence of a positive constant $C_{5}$ such that \begin{equation}\label{eq14} |u |_{q(x)}\leq C_{5}\|u \| \quad \forall u\in E. \end{equation} The last two inequalities show that there exists a positive constant $K_2(\lambda)$ such that \begin{equation}\label{eq15} \lambda \int_{\Omega}\frac{1}{q(x)}| u|^{q(x)}\,dx \leq K_2(\lambda)\left(\|u \|^{q^{-}}+\|u\|^{q^{+}} \right) \quad \forall u\in E. \end{equation} By inequalities \eqref{eq9} and \eqref{eq15}, we obtain \begin{equation}\label{eq16} J_{\lambda}(u)\leq K_1(\|u\|^{p^{-}}+\|u\|^{p^{+}} )+ K_2(\lambda)\left(\|u \|^{q^{-}}+\|u\|^{q^{+}} \right)- \frac{1}{p^{*+}}\int_{\Omega}| u|^{p^*(x)}\,dx, \end{equation} for all $u\in E$. Let $u\in E $ be arbitrary but fixed. We define $$ \Omega_{<}=\{x\in \Omega;|u(x)|<1\}. \quad \Omega_{\geq}=\Omega \backslash \Omega_{<}. $$ Then we have \begin{align*} J_{\lambda}(u) &\leq K_1(\|u\|^{p^{-}}+\|u\|^{p^{+}} )+ K_2(\lambda) \big(\|u\|^{q^{-}}+\|u\|^{q^{+}} \big)-\frac{1}{p^{*+}} \int_{\Omega}| u|^{p^*(x)}\,dx, \\ &\leq K_1(\|u\|^{p^{-}}+\|u\|^{p^{+}} )+ K_2(\lambda) \big(\|u \|^{q^{-}}+\|u\|^{q^{+}} \big) - \frac{1}{p^{*+}}\int_{\Omega_{\geq}}| u|^{p^*(x)}\,dx,\\ &\leq K_1(\|u\|^{p^{-}}+\|u\|^{p^{+}} )+ K_2(\lambda) \big(\|u \|^{q^{-}}+\|u\|^{q^{+}} \big) - \frac{1}{p^{*+}}\int_{\Omega_{\geq}}| u|^{p^{*-}}\,dx,\\ &\leq K_1(\|u\|^{p^{-}}+\|u\|^{p^{+}} )+ K_2(\lambda) \big(\|u \|^{q^{-}}+\|u\|^{q^{+}} \big) - \frac{1}{p^{*+}}\int_{\Omega}| u|^{p^{*-}}\,dx\\ &\quad +\frac{1}{p^{*+}}\int_{\Omega_{<}}| u|^{p^{*-}}\,dx. \end{align*} But there exists positive constant $K_3$ such that $$ \frac{1}{p^{*+}}\int_{\Omega_{<}}| u|^{p^{*-}}\,dx \leq K_3 \quad \forall u\in E. $$ The functional $|.|_{p^{*-}}:E\to \mathbb{R}$ defined by $$ |u|_{p^{*-}}=\Big(\int_{\Omega}|u|^{p^{*-}}dx\Big)^{1/p^{*-}} $$ is a norm in $E$. In the finite dimensional subspace $E_1$ the norm $| u|_{p^{*-}}$ and $\|u\|$ are equivalent, so there exists a positive constant $K=K(E_1)$ such that $$ \|u\| \leq K |u|_{p^{*-}} \quad \forall u\in E_1. $$ So that there exists a positive constant $K_{4}$ such that $$ J_{\lambda}(u) \leq K_1(\|u\|^{p^{-}}+\|u\|^{p^{+}} ) +K_2(\lambda)\big(\|u\|^{q^{-}}+\|u\|^{q^{+}} \big) +K_3-K_{4} \|u\|^{p^{*-}}, \quad \forall u\in E_1. $$ Hence $$ K_1(\|u\|^{p^{-}}+\|u\|^{p^{+}} ) + K_2(\lambda)\big(\|u \|^{q^{-}}+\|u\|^{q^{+}} \big) + K_3-K_{4} \|u\|^{p^{*-}}\geq 0, \quad \forall u\in S. $$ And since $p^{*-}>\max(p^{+},q^{+})$, we conclude that $S$ is bounded in $E$. \end{proof} \begin{lemma}\label{lem7} If $\{u_n\}\subset E$ is a sequence which satisfies the properties \begin{gather}\label{eq17} |J_{\lambda}(u_n)| 1$ for any integer $n$. Using \eqref{eq18} it follows that there exists $N_1>0$ such that for any $n>N_1$ we have $$ \|dJ_{\lambda}(u_n)\|\leq 1. $$ On the other hand, for all $n>N_1$ fixed, the application $E\ni v\to \langle dJ_{\lambda}(u_n),v\rangle$ is linear and continuous. The above information yield that $$ |\langle dJ_{\lambda}(u_n),v\rangle|\leq \|dJ_{\lambda}(u_n)\| \|v\| \leq \|v\| \quad \forall v\in E,\quad n>N_1. $$ Setting $v=u_n$, we have $$ -\|u_n\|\leq \int_{\Omega}|\nabla u_n|^{p(x)}\,dx - \int_{\Omega}| u_n|^{p^*(x)}\,dx +\lambda \int_{\Omega}| u_n|^{q(x)}\,dx\leq \|u_n\|, $$ for all $n>N_1$. We obtain \begin{equation}\label{eq19} -\|u_n\|- \int_{\Omega}|\nabla u_n|^{p(x)} \,dx -\lambda \int_{\Omega}| u_n|^{q(x)}\,dx\leq - \int_{\Omega}| u_n|^{p^*(x)}\,dx, \end{equation} for all $n>N_1$. Provided that $\|u_n\|>1$ relation \eqref{eq17} and \eqref{eq19} imply \begin{align*} C_6&> J_{\lambda}(u_n)\\ &\geq (\frac{1}{p^{+}}-\frac{1}{p^{*-}}) \int_{\Omega}|\nabla u_n|^{p(x)}\,dx +\lambda(\frac{1}{q^{+}}-\frac{1}{p^{*-}})\int_{\Omega} | u_n|^{q(x)}\,dx-\frac{1}{p^{*-}}\|u_n\|, \\ &\geq (\frac{1}{p^{+}}-\frac{1}{p^{*-}})\| u_n|\|^{p^-}\,dx -\frac{1}{p^{*-}}\|u_n\|. \end{align*} Letting $n\to \infty$ we obtain a contradiction. It follows that $\{u_n\}$ is bounded in $E$. And we deduce that there exists a subsequence, again denoted by $\{u_n\}$, and $u \in E $ such that $\{u_n\}$ converges weakly to $u$ in $E$. Now by Theorem \ref{thm1} we have $\nabla u_n \to \nabla u$ a.e. in $\mathbb{R}^N$ as $n\to \infty$. The proof is complete. \end{proof} \begin{proof}[Proof of Theorem \ref{thm3}] It is clear that the functional $J_{\lambda}$ is even and verifies $J_{\lambda}(0)=0$. 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