\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2016 (2016), No. 295, pp. 1--12.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2016 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2016/295\hfil Perturbed subcritical Dirichlet problems] {Perturbed subcritical Dirichlet problems with variable exponents} \author[R. Alsaedi \hfil EJDE-2016/295\hfilneg] {Ramzi Alsaedi} \address{Ramzi Alsaedi \newline Department of Mathematics, Faculty of Sciences, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia} \email{ramzialsaedi@yahoo.co.uk} \thanks{Submitted September 8, 2016. Published November 16, 2016.} \subjclass{35J60, 58E05} \keywords{Nonhomogeneous elliptic problem; variable exponent; \hfill\break\indent Dirichlet boundary condition; mountain pass theorem} \begin{abstract} We study a class of nonhomogeneous elliptic problems with Dirichlet boundary condition and involving the $p(x)$-Laplace operator and power-type nonlinear terms with variable exponent. The main results of this articles establish sufficient conditions for the existence of nontrivial weak solutions, in relationship with the values of certain real parameters. The proofs combine the Ekeland variational principle, the mountain pass theorem and energy arguments. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \allowdisplaybreaks \section{Introduction} Let $\Omega\subset\mathbb{R}$ be a bounded domain with smooth boundary. In a pioneering paper, Ambrosetti and Rabinowitz \cite{ambrab} consider the subcritical elliptic problem \begin{equation}\label{ambro} \begin{gathered} -\Delta u=|u|^{p-2}u,\quad x\in\Omega\\ u=0,\quad x\in\partial\Omega, \end{gathered} \end{equation} where $11 \text{ for all } x\in\overline\Omega\}. $$For all h\in C_+(\overline\Omega) we define$$ h^+=\sup_{x\in\Omega}h(x)\quad\text{and}\quad h^-=\inf_{x\in\Omega}h(x). $$The real numbers h^+ and h^- will play a crucial role in our arguments and usually the gap between these quantities produces new results, which are no longer valid for constant exponents. For any p\in C_+(\overline\Omega), we define the \emph{variable exponent Lebesgue space}$$ L^{p(x)}(\Omega)=\{u: u \text{ is measurable and } \int_\Omega|u(x)|^{p(x)}\,dx<\infty\}. $$This vector space is a Banach space if it is endowed with the \emph{Luxemburg norm}, which is defined by$$ |u|_{p(x)}=\inf\big\{\mu>0;\;\int_\Omega| \frac{u(x)}{\mu}|^{p(x)}\,dx\leq 1\big\}. $$Then L^{p(x)}(\Omega) is reflexive if and only if 1 < p^-\leq p^+<\infty and continuous functions with compact support are dense in L^{p(x)}(\Omega) if p^+<\infty. The inclusion between Lebesgue spaces with variable exponent generalizes the classical framework, namely if 0 <|\Omega|<\infty and p_1, p_2 are variable exponents so that p_1\leq p_2 in \Omega then there exists the continuous embedding L^{p_2(x)}(\Omega)\hookrightarrow L^{p_1(x)}(\Omega). Let L^{p'(x)}(\Omega) be the conjugate space of L^{p(x)}(\Omega), where 1/p(x)+1/p'(x)=1. For any u\in L^{p(x)}(\Omega) and v\in L^{p'(x)}(\Omega) the following H\"older-type inequality holds: \begin{equation}\label{Hol} \big|\int_\Omega uv\,dx\big|\leq\Big(\frac{1}{p^-}+ \frac{1}{p'^-}\Big)|u|_{p(x)}|v|_{p'(x)}\,. \end{equation} The \emph{modular} of L^{p(x)}(\Omega) is the mapping \rho_{p(x)}:L^{p(x)}(\Omega)\to\mathbb{R} defined by$$ \rho_{p(x)}(u)=\int_\Omega|u|^{p(x)}\,dx. $$If (u_n), u\in L^{p(x)}(\Omega) and p^+<\infty then the following relations are true: \begin{gather}\label{L4} |u|_{p(x)}>1\;\Rightarrow\;|u|_{p(x)}^{p^-}\leq\rho_{p(x)}(u) \leq|u|_{p(x)}^{p^+}, \\ \label{L5} |u|_{p(x)}<1\;\Rightarrow\;|u|_{p(x)}^{p^+}\leq \rho_{p(x)}(u)\leq|u|_{p(x)}^{p^-}, \\ \label{L6} |u_n-u|_{p(x)} \to 0\; \Leftrightarrow\;\rho_{p(x)} (u_n-u)\to 0. \end{gather} We define the variable exponent Sobolev space by$$ W^{1,p(x)}(\Omega)=\{u\in L^{p(x)}(\Omega):|\nabla u|\in L^{p(x)} (\Omega) \}. $$On W^{1,p(x)}(\Omega) we may consider one of the following equivalent norms$$ \|u\|_{p(x)}=|u|_{p(x)}+|\nabla u|_{p(x)} $$or$$\|u\|=\inf\big\{\mu>0;\;\int_\Omega\Big(| \frac{\nabla u(x)}{\mu}|^{p(x)}+|\frac{u(x)}{\mu}|^{p(x)}\Big)\,dx\leq 1\big\}\,. $$We define W_0^{1,p(x)}(\Omega) as the closure of the set of compactly supported W^{1,p(x)}-functions with respect to the norm \|u\|_{p(x)}. When smooth functions are dense, we can also use the closure of C_0^\infty(\Omega) in W^{1,p(x)}(\Omega). Using the Poincar\'e inequality, the space W_0^{1,p(x)}(\Omega) can be defined, in an equivalent manner, as the closure of C_0^\infty(\Omega) with respect to the norm$$ \|u\|_{p(x)}=|\nabla u|_{p(x)}. $$The space (W^{1,p(x)}_0(\Omega),\|\cdot\|) is a separable and reflexive Banach space. Moreover, if 0 <|\Omega|<\infty and p_1, p_2 are variable exponents so that p_1\leq p_2 in \Omega then there exists the continuous embedding W^{1,p_2(x)}_0(\Omega)\hookrightarrow W^{1,p_1(x)}_0(\Omega). Set \begin{equation}\label{rho2} \varrho_{p(x)}(u)=\int_\Omega |\nabla u(x)|^{p(x)}\,dx. \end{equation} If (u_n), u\in W^{1,p(x)}_0(\Omega) then the following properties are true: \begin{gather}\label{M4} \|u\|>1\;\Rightarrow\;\|u\|^{p^-}\leq \varrho_{p(x)}(u) \leq\|u\|^{p^+}\,, \\ \label{M5} \|u\|<1\;\Rightarrow\;\|u\|^{p^+}\leq \varrho_{p(x)}(u) \leq\|u\|^{p^-}\,, \\ \label{M6} \|u_n-u\|\to 0\;\Leftrightarrow\;\varrho_{p(x)} (u_n-u)\to 0\,. \end{gather} Set$$ p^*(x)=\begin{cases} \frac{Np(x)}{N-p(x)} &\text{if$p(x)t\}|\,dt $$has no variable exponent analogue. \item[(ii)] Variable exponent Lebesgue spaces do not have the \emph{mean continuity property}: if p is continuous and nonconstant in an open ball B, then there exists a function u\in L^{p(x)}(B) such that u(x+h)\not\in L^{p(x)}(B) for all h\in\mathbb{R}^N with arbitrary small norm. (iii) An argument in the development of the theory of function spaces with variable exponent is the fact that these spaces are never translation invariant. The use of convolution is also limited, for instance the Young inequality$$ | f*g|_{p(x)}\leq C\, | f|_{p(x)}\, \| g\|_{L^1} $$holds if and only if p is constant. \end{itemize} We refer to R\u{a}dulescu \cite{radnla} and the monographs by Diening, H\"asto, Harjulehto and Ruzicka \cite{die} and R\u{a}dulescu and Repov\v{s} \cite{radrep} for additional properties of function spaces with variable exponent and for a thorough variational analysis of these problems. \section{Main results} Let \Delta_{p(x)} denote the \emph{p(x)-Laplace operator}, namely$$ \Delta_{p(x)} u:=\operatorname{div} (|\nabla u|^{p(x)-2}\nabla u). $$In this article we study of the perturbed nonhomogeneous Dirichlet problem \begin{equation}\label{prob} \begin{gathered} -\Delta_{p(x)} u=\lambda |u|^{p(x)-2}u+|u|^{q(x)-2}u,\quad \text{in } \Omega\\ u=0,\quad \text{on } \partial\Omega. \end{gathered} \end{equation} We suppose that p,q\in C_+(\overline\Omega) satisfy \begin{equation} \label{pq} p^+\rho>0$$ \max\{\varphi(u_0),\varphi(u_1)\}<\inf[\varphi(u):\|u-u_0\|=\rho]=m_{\rho} $$and$$c=\inf_{\gamma\in\Gamma}\max_{0\leq t\leq 1}\varphi(\gamma(t))\quad \text{with}\quad \Gamma=\{\gamma\in C([0,1],X):\gamma(0)=u_0,\gamma(1)=u_1\}. $$Then c\geq m_{\rho} and c is a critical value of \varphi. \end{theorem} As pointed out by Brezis and Browder \cite{brebro}, the mountain pass theorem extends ideas already present in Poincar\'e and Birkhoff". More generally, this result is in fact true in Banach-Finsler manifolds. Assumption \eqref{pq} guarantees that the energy functional associated with \eqref{prob} has a mountain pass geometry. We study in what follows a related perturbed problem, provided that Theorem \ref{th1} cannot be applied. Consider the problem \begin{equation}\label{prob1} \begin{gathered} -\Delta_{p(x)} u=\lambda |u|^{p(x)-2}u+\mu |u|^{q(x)-2}u, \quad \text{in } \Omega\\ u=0, \quad \text{on } \partial\Omega. \end{gathered} \end{equation} We say that u is a \emph{weak solution} of problem \eqref{prob1} if u\in W^{1,p(x)}_0(\Omega)\setminus\{0\} and$$ \int_\Omega \big( |\nabla u|^{p(x)-2}\nabla u\cdot\nabla v-\lambda|u|^{p(x)-2} u v \big)dx -\mu\int_\Omega |u|^{q(x)-2}uv\,dx=0, $$for all v\in W^{1,p(x)}_0(\Omega). We assume that p,q\in C_+(\overline\Omega) satisfy \begin{equation} \label{pq1} q^-0 such that for all \mu<\mu^* problem \eqref{prob1} has at least one solution. \end{theorem} The key ingredient in the proof of Theorem \ref{t2} is the Ekeland variational principle, which asserts the existence of almost critical points of \mathcal{E}. The subcritical framework of the problem yields a nontrivial critical point of \mathcal{E}, hence a weak solution of problem \eqref{prob1}. We point out that the Ekeland variational principle can be viewed as the nonlinear version of the Bishop-Phelps theorem \cite{bis1}. The arguments developed in this paper show that a similar result holds if the p(x)-Laplace operator is replaced with other nonhomogeneous differential operators with variable exponent, for instance the \emph{generalized mean curvature operator} defined by$$ \operatorname{div} \Big((1+|\nabla u|^2)^{[p(x)-2]/2}\nabla u \Big). $$\section{Proof of Theorem \ref{t1}} The proof strongly relies on the mountain pass theorem in relationship with some ideas developed in \cite{mora} and \cite{pucci}. We start with the verification of the geometric hypotheses of the mountain pass theorem. We observe that \mathcal{E}(0)=0 and we show the existence of a mountain near the origin, namely there exist positive numbers r and \eta such that \mathcal{E}(u)\geq\eta for all u\in W^{1,p(x)}_0(\Omega) with \|u\|=r. We first observe that the definition of \lambda^* combined with the fact that \lambda<\lambda^* imply that there exists \delta>0 such that$$ \int_\Omega\frac{1}{p(x)}(|\nabla u|^{p(x)}-\lambda |u|^{p(x)})dx \geq \delta |\nabla u|_{p(x)},\quad\text{for all}\ u\in W^{1,p(x)}_0(\Omega). $$But$$ \int_\Omega\frac{|u|^{q(x)}}{q(x)}dx\leq\frac{1}{q^-}|u|_{q(x)}, \quad\text{for all } u\in W^{1,p(x)}_0(\Omega). $$Combining these inequalities, we deduce that \begin{equation} \label{enough} \mathcal{E}(u)\geq \delta\,|\nabla u|_{p(x)}-\frac{1}{q^-}\,|u|_{q(x)}\,. \end{equation} Fix r\in(0,1) and u\in W^{1,p(x)}_0(\Omega) with \|u\|=r. Then relations \eqref{M5}, \eqref{enough} and the Sobolev embedding W^{1,p(x)}_0(\Omega)\hookrightarrow L^{q(x)}(\Omega) yield$$ \mathcal{E}(u)\geq\delta\,\|u\|^{p^+}-\frac{C}{q^-}\,\|u\|^{q^-}. Choosing eventually r\in(0,1) smaller if necessary, we conclude that there exists \eta>0 such that \mathcal{E}(u)\geq\eta for all u\in W^{1,p(x)}_0(\Omega) with \|u\|=r. Next, we argue the existence of a valley over the chain of mountains. For this purpose, we fix s>1 and w\in W^{1,p(x)}_0(\Omega)\setminus\{0\}. It follows that \begin{equation} \label{abcc} \begin{aligned} \mathcal{E} (sw) & =\int_\Omega \frac{s^{p(x)}}{p(x)}\left( |\nabla w|^{p(x)} -\lambda |w|^{q(x)}\right)dx-\int_\Omega\frac{s^{q(x)}}{q(x)}\,|w|^{q(x)}dx\\ &\leq A \frac{s^{p^+}}{p^-}-B \frac{s^{q^-}}{q^+}, \end{aligned} \end{equation} where A=\int_\Omega \left( |\nabla w|^{p(x)}-\lambda |w|^{q(x)}\right)dx\quad\text{and}\quad B=\int_\Omega |w|^{q(x)}dx. Using hypothesis \eqref{pq}, relation \eqref{abcc} yields \mathcal{E} (sw)<0 for s large enough. To apply Theorem \ref{th1} to our problem \eqref{prob} it remains to check that the energy functional \mathcal{E} satisfies the Palais-Smale compactness condition. Let (u_n)\subset W^{1,p(x)}_0(\Omega) be an arbitrary Palais-Smale sequence for \mathcal{E}, namely \begin{equation}\label{ps1} \mathcal{E} (u_n)=O(1)\quad\text{as } n\to\infty \end{equation} and \begin{equation} \label{ps2} \|\mathcal{E}' (u_n)\|_{W^{-1,p'(x)}(\Omega)}=o(1)\quad\text{as } n\to\infty\,. \end{equation} We claim that \begin{equation} \label{claim} \text{the sequence (u_n) is bounded in } W^{1,p(x)}_0(\Omega) . \end{equation} Relations \eqref{ps1} and \eqref{ps2} yield \begin{equation} \label{ps3} \int_\Omega\frac{1}{p(x)}\Big( |\nabla u_n|^{p(x)}-\lambda |u_n|^{p(x)}\Big)dx -\int_\Omega\frac{1}{q(x)} |u_n|^{q(x)}dx=O(1)\quad\text{as } n\to\infty \end{equation} and for all v\in W^{1,p(x)}_0(\Omega), \begin{equation} \label{ps4} \begin{aligned} &\int_\Omega\left( |\nabla u_n|^{p(x)-2}\nabla u_n\cdot\nabla v -\lambda |u_n|^{p(x)-2}u_nv\right)dx-\int_\Omega |u_n|^{q(x)-2}u_nv\,dx \\ &= o(1)\|v\|\quad\text{as } n\to\infty\,. \end{aligned} \end{equation} Choosing v=u_n in \eqref{ps4} we deduce that \begin{equation} \label{ps5} \int_\Omega\Big( |\nabla u_n|^{p(x)}-\lambda |u_n|^{p(x)}\Big)dx -\int_\Omega |u_n|^{q(x)}dx=o(1)\|u_n\|\quad\text{as } n\to\infty\,. \end{equation} On the other hand, relation \eqref{ps3} implies \begin{align*} & O(1)+\frac{1}{p^+}\int_\Omega\big( |\nabla u_n|^{p(x)}-\lambda |u_n|^{p(x)}\big)dx \\ &\leq \int_\Omega\frac{1}{q(x)}\, |u_n|^{q(x)}dx \\ &\leq O(1)+\frac{1}{p^-}\int_\Omega\big( |\nabla u_n|^{p(x)} -\lambda |u_n|^{p(x)}\big)dx. \end{align*} Using now relation \eqref{ps5} we deduce that \begin{align*} & O(1)+o(1)\|u_n\|+\frac{1}{p^+}\int_\Omega |u_n|^{q(x)}dx \\ &\leq \int_\Omega\frac{1}{q(x)}\, |u_n|^{q(x)}dx \\ &\leq O(1)+o(1)\|u_n\|+\frac{1}{p^-}\int_\Omega |u_n|^{q(x)}dx. \end{align*} It follows that \begin{equation} \label{ps6} \int_\Omega |u_n|^{q(x)}dx=O(1)+o(1)\|u_n\|\quad\text{as } n\to\infty\,. \end{equation} Returning to \eqref{ps3} and using relation \eqref{ps6} we deduce that \begin{equation} \label{ps7} \int_\Omega\frac{1}{p(x)}\big( |\nabla u_n|^{p(x)}-\lambda |u_n|^{p(x)}\big)dx =O(1)+o(1)\|u_n\|\,. \end{equation} Taking into account the definition of \lambda^* and the fact that \lambda<\lambda^*, relation \eqref{ps7} implies that (u_n) is bounded in W^{1,p(x)}_0(\Omega), hence Claim \eqref{claim} is argued. So, up to a subsequence \begin{gather} \label{cow} u_n\rightharpoonup u\quad\text{in } W^{1,p(x)}_0(\Omega), \\ \label{col} u_n\to u\quad\text{in}\ L^{p(x)}(\Omega). \end{gather} We prove in what follows that \begin{equation} \label{cra0}(u_n) \text{ contains a strongly convergent subsequence in W^{1,p(x)}_0(\Omega)}. \end{equation} We first observe that relation \eqref{ps4} yields \begin{equation} \label{cra1} \int_\Omega |\nabla u_n|^{p(x)-2}\nabla u_n\cdot\nabla v\,dx =\int_\Omega \varrho(x,u_n)v\,dx + o(1)\|v\|\quad\text{as}\ n\to\infty\,, \end{equation} for all v\in W^{1,p(x)}_0(\Omega) where \varrho (x,w)=\lambda |w|^{p(x)-2}w+|w|^{q(x)-2}w. We assume in what follows that p^+0 there is a subset A of \Omega of measure less than \eta and such that u_n\to u\quad\text{uniformly on}\ \Omega\setminus A. $$So, our claim \eqref{cra2} follows as soon as we show that for for all \varepsilon>0 and n large enough \begin{equation} \label{cra3} \int_A|\varrho (x,u_n)-\varrho (x,u)|^{Np(x)/[Np(x)-N+p(x)]}dx\leq\varepsilon\,. \end{equation} Taking into account the subcritical growth of \varrho we have for some C>0$$ \int_A|\varrho (x,u)|^{Np(x)/[Np(x)-N+p(x)]}dx \leq C \int_A(1+|u|)^{Np(x)/[N-p(x)]}dx, which can be made sufficiently small if we choose \eta>0 small enough. We conclude that \begin{align*} &\int_A|\varrho (x,u_n)-\varrho (x,u)|^{Np(x)/[Np(x)-N+p(x)]}dx \\ &\leq \delta \int_A|u_n-u|^{Np(x)/[N-p(x)]}dx+C_\delta |A|<\varepsilon, \end{align*} by choosing \eta small enough and by using Sobolev embeddings and \eqref{claim}. Our claim \eqref{cra0} is proved, which implies that \mathcal{E} satisfies the Palais-Smale condition. Applying Theorem \ref{th1} we deduce that \eqref{prob} has a nontrivial solution for all \lambda<\lambda^*. The ideas developed in the proof of Theorem \ref{t1} allow to extend this result for more general perturbations. Indeed, let us consider the nonhomogeneous Dirichlet problem: \begin{equation}\label{probbis} \begin{gathered} -\Delta_{p(x)} u+a(x) |u|^{p(x)-2}u=|u|^{q(x)-2}u,\quad \text{in } \Omega\\ u=0,\quad \text{on } \partial\Omega, \end{gathered} \end{equation} where a\in L^\infty(\Omega) and there exists \alpha>0 such that \begin{equation} \label{pq11} \int_\Omega\left(|\nabla u|^{p(x)}+a(x)|u|^{p(x)}\right) \geq\alpha\rho_{p(x)}(\nabla u)\quad\text{for all}\ u\in W^{1,p(x)}_0(\Omega). \end{equation} The following property is the counterpart of \cite[Theorem 10]{pucci} in the framework of operators and nonlinearities involving variable exponents. \begin{theorem}\label{t1bis} Suppose that hypotheses \eqref{pq} and \eqref{pq11} are satisfied. Then \eqref{probbis} has at least one nontrivial solution. \end{theorem} \section{Proof of Theorem \ref{t2}} The weak solutions of \eqref{prob1} correspond to critical points of the associated energy functional \mathcal{J}:W^{1,p(x)}_0(\Omega)\to\mathbb{R} defined by \mathcal{J}(u)=\int_\Omega\frac{1}{p(x)}(|\nabla u|^{p(x)}-\lambda |u|^{p(x)})dx -\mu\int_\Omega\frac{|u|^{q(x)}}{q(x)}dx, for all  u\in W^{1,p(x)}_0(\Omega). We first establish a preliminary result, which asserts that the energy functional \mathcal{J} satisfies the geometric condition around the origin, provided that \lambda<\lambda^* and \mu is small enough. \begin{lemma}\label{lema1} Let \lambda<\lambda^*. Then there exist positive numbers \mu^*, r and a such that for all \mu<\mu^* we have \mathcal{J}(u)\geq a, provided that u\in W^{1,p(x)}_0(\Omega) satisfies \|u\|=r. \end{lemma} \begin{proof} As in the proof of Theorem \ref{t1}, combining the definition of \lambda^* with the fact that \lambda<\lambda^*, we deduce that there is a positive number \delta such that \begin{equation} \label{amiens1} \mathcal{J}(u)\geq \delta |\nabla u|_{p(x)}-\frac{\mu}{q^-}\,|u|_{q(x)}\quad \text{for all u\in W^{1,p(x)}_0(\Omega)}. \end{equation} Let a_1>0 denote the best constant corresponding to the continuous embedding of W^{1,p(x)}_0(\Omega) into L^{q(x)}(\Omega). Fix arbitrarily r\in (0,1). Using \eqref{L5} we obtain \begin{equation} \label{amiens2} \int_\Omega |u|^{q(x)}dx\leq |u|^{q^-}_{q(x)}\leq a_1^{q^-}\|u\|^{q^-} \quad\text{for all u\in W^{1,p(x)}_0(\Omega) }, \|u\|=\frac{r}{a_1}. \end{equation} Next, using \eqref{M5} we obtain \begin{equation} \label{amiens3} |\nabla u|_{p(x)}\geq \|u\|^{p^+}\quad\text{for all u\in W^{1,p(x)}_0(\Omega)},\quad \|u\|=\frac{r}{a_1}. \end{equation} We notice that in \eqref{amiens2} and \eqref{amiens3} we can assume that a_1 is large enough in order to have r/a_1<1. Relations \eqref{amiens1}, \eqref{amiens2} and \eqref{amiens3} imply that for all u\in W^{1,p(x)}_0(\Omega) with \|u\|=\frac{r}{a_1} we have \begin{align*} \mathcal{J}(u) &\geq\delta\|u\|^{p^+}-\frac{\mu}{q^-}\,a_1^{q^-}\|u\|^{q^-}\\ & =\delta r^{p^+}-\frac{\mu}{q^-}\, a_1^{q^-}r^{q^-}\\ &=r^{q^-}\Big( \delta r^{p^+-q^-}-\frac{\mu}{q^-}\,a_1^{q^-}\Big). \end{align*} This relation shows that the lemma follows after choosing \mu^*=\frac{\delta q^-r^{p^+-q^-}}{2a_1^{q^-}}\quad\text{and}\quad a=\frac{\delta r^{p^+}}{2}\,. $$\end{proof} The second auxiliary result shows that \mathcal{J} has a valley around the origin, hence the hypotheses of the mountain pass theorem are not fulfilled. \begin{lemma}\label{lema2} Let \lambda<\lambda^*. Then there exists a smooth nonnegative function \phi\in W^{1,p(x)}_0(\Omega) such that \mathcal{J}(t\phi)<0 for all t>0 sufficiently small. \end{lemma} \begin{proof} The basic idea is to use our assumption \eqref{pq1}, more exactly q^-0\quad\text{and}\quad B:=\int_{\omega} \phi^{q(x)}dx>0.$$ We deduce that $\mathcal{J}(t\phi)<0$, provided that $t\in(0,1)$ is small enough. \end{proof} Fix $\mu^*>0$ as established in Lemma \ref{lema1} and let $\mu<\mu^*$. Using Lemmata \ref{lema1} and \ref{lema2} we deduce that there exists $r>0$ such that $$\inf_{u\in\overline{B_r(0)}}\mathcal{J} (u)<0<\inf_{u\in\partial B_r(0)}\mathcal{J} (u),$$ where $B_r(0):=\{u\in W^{1,p(x)}_0(\Omega); \|u\|0$ such that $$\varepsilon<\inf_{u\in\partial B_r(0)}\mathcal{J} (u)-\inf_{u\in\overline{B_r(0)}}\mathcal{J} (u).$$ Applying Ekeland's variational principle \cite{eke} we find $u_\varepsilon\in \overline{B_r(0)}$ such that \begin{gather} \label{ivar1} \mathcal{J} (u_\varepsilon)<\inf_{u\in\overline{B_r(0)}}\mathcal{J} (u)+\varepsilon, \\ \label{ivar2} \mathcal{J} (u_\varepsilon)<\mathcal{J} (u)+\varepsilon \|u-u_\varepsilon\|\quad\text{for all $u\in W^{1,p(x)}_0(\Omega)\setminus\{u_\varepsilon\}$}. \end{gather} We claim that $u_\varepsilon\in B_r(0)$. Indeed, relations \eqref{ivar1} and \eqref{ivar2} yield $$J(u_\varepsilon)<\inf_{u\in\overline{B_r(0)}}\mathcal{J} (u)+\varepsilon<\inf_{u\in\partial B_r(0)}\mathcal{J} (u),$$ hence $u_\varepsilon\not\in\partial B_r(0)$. A standard argument (see, e.g., \cite[p. 2934]{mrpams}) shows that $\|\mathcal{J}'(u_\varepsilon)\|\leq\varepsilon$. Let $c:=\inf_{u\in\overline{B_r(0)}}\mathcal{J} (u)$. It follows that $u_\varepsilon$ is an \emph{almost critical point} of $\mathcal{J}$ at the level $c$, that is, $$\lim_{\varepsilon\to 0}\mathcal{J} (u_{\varepsilon})=c\quad\text{and}\quad \lim_{\varepsilon\to 0}\|\mathcal{J}' (u_{\varepsilon})\|=0.$$ From now on, using the same argument as in \cite[p. 59]{gazrad} (see also \cite[Theorem 3.1]{fan} and \cite[p. 2935]{mrpams}) we deduce that there exists a subsequence $(v_n)$ of $(u_\varepsilon)$ that converges to a nontrivial critical point $u$ of $\mathcal{J}$, hence $u$ is a weak solution of problem \eqref{prob1}. \section*{Perspectives and open problems} Now we raise some open problems in relationship with the study developed in this paper. \smallskip \noindent\textbf{Open problem 1.} Problem \eqref{prob} has been studied in the subcritical case, namely under the basic hypothesis \eqref{pq}, namely \$\max\{p(x),q(x)\}