\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 176, pp. 1--14.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2013 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2013/176\hfil Existence of solutions] {Existence of solutions for eigenvalue problems with nonstandard growth conditions} \author[S. Aouaoui \hfil EJDE-2013/176\hfilneg] {Sami Aouaoui} % in alphabetical order \address{Sami Aouaoui \newline D\'epartement de math\'ematiques, Facult\'e des sciences de Monastir \newline Rue de l'environnement, 5019 Monastir, Tunisia} \email{aouaouisami@yahoo.fr, Phone +216 73500216, Fax +216 73500218} \thanks{Submitted April 22, 2013. Published July 30, 2013.} \subjclass[2000]{35D30, 35J20, 35J62, 58E05} \keywords{Critical point; energy functional; eigenvalue problem; \hfill\break\indent variable exponent; Ekeland's lemma} \begin{abstract} We prove the existence of weak solutions for some eigenvalue problems involving variable exponents. Our main tool is critical point theory. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction and statement of main results} In this article, we are concerned with the quasilinear problem \begin{equation} \label{ePl} -\operatorname{div}(| \nabla u|^{p(x)-2} \nabla u) + |u|^{p(x)-2} u = \lambda \varphi(x) |u|^{ \alpha(x)-2} u + h,\quad \text{in }\mathbb{R}^N, \end{equation} where $ N \geq 3$, $p $ and $ \alpha \in \{v \in C( \mathbb{R}^N, \mathbb{R}) \cap L^{ \infty}( \mathbb{R}^N), \inf_{x \in \mathbb{R}^N} v(x) > 1 \}$, $\varphi \in C( \mathbb{R}^N, \mathbb{R})$, $\varphi(x) > 0$ for all $x \in \mathbb{R}^N$, $\lambda $ is a positive parameter and $ h$ is a function which belongs to the dual of the Sobolev space with variable exponent $ W^{1, p( \cdot)}( \mathbb{R}^N)$. The study of eigenvalue problems involving variable exponents growth conditions has been an interesting topic of research in last years. We can for example refer to \cite{f1,f4,m1,m2,m3,m4,m5}. A first contribution in this sense is due to Fan, Zhand and Zhao \cite{f4} who studied the problem \begin{equation} \label{e1} \begin{gathered} - \operatorname{div}(| \nabla u|^{p(x)-2} \nabla u) = \lambda |u|^{p(x)-2} u \quad \text{in }\Omega \\ u = 0 \quad \text{on }\partial \Omega, \end{gathered} \end{equation} where $ \Omega \subset \mathbb{R}^N $ is a bounded domain with smooth boundary$, p : \overline{ \Omega} \to (1, \infty) $ is a continuous function and $ \lambda $ is a real number. In \cite{f4}, the authors established the existence of infinitely many eigenvalues for problem \eqref{e1}. Denoting $ \Lambda $ the set of all nonnegative eigenvalues, it was proved in \cite{f4} that $ \sup ( \Lambda) = + \infty$. It was also proved that only under special conditions concerning the monotony of the variable exponent $ p ( \cdot)$, we have $ \inf( \Lambda) > 0 $ which is in contrast with the case when $ p $ is a constant. Mih\v ailescu and R\v adulescu \cite{m2} studied the problem \begin{equation} \label{e2} \begin{gathered} - \operatorname{div}(| \nabla u|^{p(x)-2} \nabla u) = \lambda |u|^{q(x)-2} u \quad \text{in }\Omega \\ u = 0 \quad \text{on }\partial \Omega, \end{gathered} \end{equation} where $ \Omega \subset \mathbb{R}^N $ is a bounded domain with smooth boundary$, p, q : \overline{ \Omega} \to ( 1, + \infty) $ are two continuous functions and $ \lambda $ is a real number. Using Ekeland's variational principle, they proved that under the assumption $$ \min_{x \in \overline{ \Omega}} q(x) < \min_{x \in \overline{ \Omega}} q(x) < \max_{x \in \overline{ \Omega}} q(x),\quad \max_{x \in \overline{ \Omega}} q(x) < N, \quad q(x) < \frac{ N p(x)}{ N -p(x)}\quad \forall x \in \overline{ \Omega}, $$ there exists a continuous family of eigenvalues which lies in a neighborhood of the origin. The case when $ \max_{x \in \overline{ \Omega}} p(x) < \min_{x \in \overline{ \Omega}} q(x)$ was treated by Fan and Zhang \cite{f3} using the Mountain-Pass Theorem. Finally, in the case when $ \max_{x \in \overline{ \Omega}} p(x) < \min_{x \in \overline{ \Omega}} q(x)$ and by combining results of \cite{f3} and \cite{m3}, it is easy to see that there exists two positive constants $ \lambda^* $ and $ \lambda^{**} $ such that any $ \lambda \in (0, \lambda^*) \cup ( \lambda^{**}, + \infty) $ is an eigenvalue of the problem. Another important eigenvalue problem is the following \begin{equation} \label{e3} \begin{gathered} - \operatorname{div}((| \nabla u|^{p_1(x)-2} + | \nabla u|^{p_2(x)-2}) \nabla u) = \lambda |u|^{q(x)-2} u \quad \text{in }\Omega \\ u = 0 \quad \text{on }\partial \Omega, \end{gathered} \end{equation} where $ \Omega \subset \mathbb{R}^N $ is a bounded domain with smooth boundary. Provided that $ p_1, p_2, q : \overline{ \Omega} \to ( 1, + \infty) $ are continuous functions such that $ q $ has a sub-critical growth with respect to $ p_2 $ and the following condition is verified $$ 1 < p_2(x) < \min_{ \overline{ \Omega}} q \leq \max_{ \overline{ \Omega}} q < p_1(x)\quad \forall x \in \overline{ \Omega}, $$ problem \eqref{e3} was discussed in \cite{m4} and it was shown that there exist two positive constants $ \lambda_0 $ and $ \lambda_1 $ with $ \lambda_0 \leq \lambda_1 $ such that any $ \lambda \in [ \lambda_1, + \infty) $ is an eigenvalue of the problem \eqref{e3} while for any $ \lambda \in (0, \lambda_0)$, problem \eqref{e3} does not admit any nontrivial solution. The novelty in this article lies in the fact that we divide $ \mathbb{R}^N $ into three parts \begin{gather*} \Omega_1 = \{x \in \mathbb{R}^N: \alpha(x) < p(x) \},\quad \Omega_2 = \{x \in \mathbb{R}^N: \alpha(x) > p(x) \},\\ \Omega_3 = \{x \in \mathbb{R}^N: \alpha(x) = p(x) \}. \end{gather*} We assume that $ \operatorname{meas}( \Omega_3) = 0 $ where ``meas'' denotes the Lebesgue measure in $ \mathbb{R}^N$. In this work, we are interested in the case when $ \operatorname{meas}(\Omega_1) > 0 $ and $ \operatorname{meas}(\Omega_2) > 0$. Thus, possibly we could have $ \operatorname{meas}(\Omega_1)= + \infty $ and $ \operatorname{meas}(\Omega_2) = + \infty$. We have to notice that this possibility to divide $ \mathbb{R}^N $ into $ \Omega_1, \Omega_2 $ and $ \Omega_3 $ is so related to quasilinear equations involving variable exponents because we cannot find such a phenomenon when treating quasilinear equations with constant exponents. On the other hand, in the majority of works dealing with nonlinear equations involving variable exponents, a precise comparison between the extrema of involved variable exponents is provided. So, the situation that we are treating is rather new. Throughout this paper, we denote \begin{gather*} \alpha_{ \Omega_1}^- = \inf_{x \in \Omega_1} \alpha(x),\quad \alpha_{ \Omega_2}^- = \inf_{x \in \Omega_2} \alpha(x), \\ p_{ \Omega_1}^- = \inf_{x \in \Omega_1} p(x),\quad p_{ \Omega_1}^+= \sup_{x \in \Omega_1} p(x), \\ p_{ \Omega_2}^- = \inf_{x \in \Omega_2} p(x),\quad p_{ \Omega_2}^+= \sup_{x \in \Omega_2} p(x), \end{gather*} $p^+ = \sup_{x \in \mathbb{R}^N} p(x)$, $\|h\|_{-1}$ is the norm of $h$ in the dual of $W^{1, p ( \cdot)}( \mathbb{R}^N)$. Set $$ E = \big\{u \in W^{1, p( \cdot)}( \mathbb{R}^N), \int_{\mathbb{R}^N} \varphi(x) |u|^{ \alpha(x)} dx < + \infty \big\}. $$ We equip the functional space $ E $ with the norm $$ \|u\|_E = \|u\|_{W^{1, p( \cdot)}( \mathbb{R}^N)} + | (\varphi( \cdot))^{ \frac{1}{ \alpha(\cdot)}} u|_{L^{ \alpha(\cdot)}( \mathbb{R}^N)}. $$ \noindent\textbf{Definition} A function $ u \in E $ is said to be a weak solution of the problem \eqref{ePl} if it satisfies \begin{align*} & \int_{ \mathbb{R}^N} | \nabla u|^{p(x)-2} \nabla u \nabla v dx + \int_{ \mathbb{R}^N} |u|^{p(x)-2} u v dx \\ & = \lambda \int_{ \mathbb{R}^N} \varphi(x) |u|^{ \alpha(x) -2} uv dx + \int_{ \mathbb{R}^N} h v dx,\quad \forall v \in E. \end{align*} \smallskip This article is divided into two parts. In the first part, we will study problem \eqref{ePl} under the following hypotheses: \begin{itemize} \item[(H1)] $\int_{ \Omega_1} ( \varphi(x))^{ \frac{p(x)}{ p(x) - \alpha(x)}} dx < + \infty$; \item[(H2)] $ p(x) < N$ for all $x \in \Omega_2 $, and there exists $ r \in C_+( \overline{ \Omega_2}) $ such that $ \varphi \in L^{r( \cdot)}( \Omega_2) $ and $$ p(x) \leq \frac{ \alpha(x) r(x)}{r(x) -1} \leq p^*(x)\quad \forall x \in \Omega_2,\quad \text{where } p^*(x) = \frac{Np(x)}{N -p(x)}; $$ \item[(H3)] There exists $ \psi \in W^{1, p( \cdot)}( \mathbb{R}^N) $ such that $ \int_{ \mathbb{R}^N} h(x) \psi(x) > 0$. \end{itemize} The main result of this first part is given by the following theorem. \begin{theorem} \label{thm1.1} Assume that {(H1), (H2)} hold. Assume also that $ \alpha_{ \Omega_2}^- \geq p_{ \Omega_2}^+$. Then, we have: if {\rm (H3)} holds, or $ h = 0$, then there exists $ \lambda_* > 0 $ such that for all $ 0 < \lambda < \lambda_*$, there exists $ \eta_{ \lambda} > $ verifying that: if $ \|h\|_{-1} < \eta_{ \lambda}$, then problem \eqref{ePl} admits at least one nontrivial weak solution $ u_{0, \lambda}$. Moreover, if $ h = 0$, then $ u_{0, \lambda} \to 0 $ strongly in $ W^{1, p( \cdot)}( \mathbb{R}^N) $ when $ \lambda \to 0$. \end{theorem} In the second part of this article, we will remove the assumptions (H1) and (H2) and we will study \eqref{ePl} under the following hypotheses: \begin{itemize} \item[(H4)] The exponent $ p( \cdot) $ is log-H\"older continuous; i.e., there exists a positive constant $ D > 0 $ such that $$ |p(x) -p(y)| \leq \frac{D}{-\text{log} (|x-y|)},\quad \text{for every $ x, y \in \mathbb{R}^N$ with $|x-y| \leq 1/2$}; $$ \item[(H5)] $\inf_{x \in \mathbb{R}^N} \alpha(x) = \alpha^- > 2 $. \end{itemize} \begin{theorem} \label{thm1.2} Assume that {\rm (H4), (H5)} hold. If $ h = 0$, then there exists $ 0 < \lambda_{**} $ such that for every $ 0 < \lambda < \lambda_{**}$, then problem \eqref{ePl} admits at least one nontrivial weak solution. \end{theorem} \begin{remark} \label{rmk1.1}\rm The importance of the hypothesis (H4) lies in the fact that if $ p $ verifies the logarithmic H\"older continuity condition (also called the Dini-Lipschitz condition), the space $ C_0^{ \infty}( \mathbb{R}^N) $ is dense in $ W^{1, p (\cdot)}( \mathbb{R}^N) $ (see \cite{e2,s1}). \end{remark} \section{Preliminaries} First, we give some background facts from the variable exponent Lebesgue and Sobolev spaces. For details, we refer to the books \cite{d1,m6} and the papers \cite{e1,f2,h1,s2}. Assume $ \Omega \subset \mathbb{R}^N $ is a (bounded or unbounded) open domain. Set $ C_+( \overline{ \Omega})= \{h \in C( \overline{ \Omega}) \cap L^{ \infty}( \Omega),\, h(x) > 1,\, \forall x \in \overline{\Omega}\}$. For any $ p \in C_+( \overline{ \Omega})$, we define $$ p^+ = \sup_{ x \in \Omega} p(x)\quad \text{and } p^- = \inf_{x \in \Omega} p(x). $$ For each $ p \in C_+( \overline{ \Omega})$, we define the variable exponent Lebesgue space $$ L^{p(\cdot)}( \Omega) = \{u;\ u: \Omega \to \mathbb{R} \text{ measurable such that } \int_{ \Omega}|u(x)|^{p(x)}dx < + \infty \}. $$ This space becomes a Banach space with respect to the Luxemburg norm, $$ |u|_{L^{p(\cdot)}( \Omega)} = \inf \{ \mu > 0: \int_{ \Omega}| \frac{u(x)}{ \mu}|^{p(x)} dx \leq 1\}. $$ Moreover$, L^{p(\cdot)}( \Omega) $ is a reflexive space provided that $ 1 < p^- \leq p^+ < + \infty$. Denoting by $ L^{p'(\cdot)}( \Omega) $ the conjugate space of $ L^{p(\cdot)}( \Omega) $ where $ \frac{1}{p(x)} + \frac{1}{p'(x)} = 1$; for any $ u \in L^{p(\cdot)}( \Omega) $ and $ v \in L^{p'(\cdot)}( \Omega) $ we have the following H\"older type inequality \begin{equation} | \int_{ \Omega} u v dx| \leq 2 |u|_{L^{p(\cdot)}( \Omega)}|v|_{L^{p'(\cdot)} ( \Omega)}. \label{e2.1} \end{equation} Now, we introduce the modular of the Lebesgue-Sobolev space $ L^{p(\cdot)}( \Omega) $ as the mapping $ \rho_{p(\cdot)}: L^{p(\cdot)}( \Omega) \to \mathbb{R} $ defined by $$ \rho_{p(\cdot)}(u) = \int_{ \Omega}|u|^{p(x)}dx,\quad u \in L^{p( \cdot)}( \Omega). $$ Here, we give some relations which could be established between the Luxemburg norm and the modular. If $ (u_n)_n, u \in L^{p(\cdot)}( \Omega) $ and $ 1 < p^- \leq p^+ < + \infty$, then the following relations hold: \begin{gather} |u|_{L^{p(\cdot)}( \Omega)} > 1 \Rightarrow\ |u|_{L^{p(\cdot)}(\Omega)}^{p^-} \leq \rho_{p(\cdot)}(u) \leq |u|_{L^{p(\cdot)}(\Omega)}^{p^+}, \label{e2.2} \\ |u|_{L^{p(\cdot)}( \Omega)} < 1 \Rightarrow\ |u|_{L^{p(\cdot)}( \Omega)}^{p^+} \leq \rho_{p(\cdot)}(u) \leq |u|_{L^{p(\cdot)}( \Omega)}^{p^-}, \label{e2.3} \\ |u_n - u|_{L^{p(\cdot)}( \Omega)} \to 0 \Leftrightarrow \rho_{p(\cdot)}(u_n-u) \to 0. \label{e2.4} \end{gather} Next, we define $ W^{1,p(\cdot)}( \Omega) $ as the space $$ W^{1,p(\cdot)}( \Omega) = \{u \in L^{p(\cdot)}( \Omega): | \nabla u| \in L^{p(\cdot)}(\Omega) \} $$ and it can be equipped with the norm $ \|u\|_{1,p(\cdot)}= |u|_{L^{p(\cdot)}(\Omega)} + | \nabla u|_{L^{p(\cdot)}( \Omega)}$. The space $ W^{1,p(\cdot)}( \Omega) $ is a Banach space which is reflexive under condition $ 1 < p^- \leq p^+ < + \infty$. Let $ p, q \in C_+( \overline{\Omega})$. If we have $ p(x) \leq q(x) \leq p^{*}(x)$ for all $x \in \overline{ \Omega}$, where \[ (p^*(x) = \begin{cases} \frac{Np(x)}{N-p(x)} &\text{if } p(x) < N,\\ \infty &\text{if } p(x) \geq N; \end{cases} \] then there is a continuous embedding $ W^{1,p(\cdot)}( \Omega) \hookrightarrow L^{q(\cdot)}( \Omega)$. This last embedding is compact provided that $ \Omega $ is bounded in $ \mathbb{R}^N $ and that $ q(x) < p^*(x)$ for all $x \in \overline{ \Omega}$. \section{Proof of Theorem \ref{thm1.1}} Here, we notice that since $ \alpha( \cdot) $ satisfies the conditions (H1) and (H2), it is easy to see that $ E = W^{1, p ( \cdot)}( \mathbb{R}^N)$. In this first part, we will equip $ E $ with the norm $$ \|u\| = \|u\|_{W^{1, p( \cdot)}( \Omega_1)} + \|u\|_{ W^{1, p( \cdot)}( \Omega_2)} $$ which is clearly equivalent to the norm $ \|\cdot\|_E $ or $ \|\cdot \|_{W^{1, p ( \cdot)}( \mathbb{R}^N)}$. Let $ J_{ \lambda} : W^{1, p( \cdot)}( \mathbb{R}^N) \to \mathbb{R} $ be the energy functional given by \[ J_{ \lambda} (u) = \int_{ \mathbb{R}^N} \frac{| \nabla u|^{p(x)} + |u|^{p(x)}}{p(x)} dx - \lambda \int_{ \mathbb{R}^N} \frac{\varphi(x)}{ \alpha(x)} |u|^{ \alpha(x)} dx - \int_{ \mathbb{R}^N} h u dx. \] Using inequality \eqref{e2.1} and hypotheses (H1) and (H2), it is easy to see that the functional $ J_{ \lambda} $ is well defined on $ W^{1, p ( \cdot)}( \mathbb{R}^N)$. Moreover, by classical arguments we have that $ J_{ \lambda} \in C^1( W^{1, p( \cdot)}( \mathbb{R}^N), \mathbb{R}) $ and \begin{align*} \langle J'_{ \lambda}(u), v\rangle & = \int_{ \mathbb{R}^N} | \nabla u|^{p(x)-2} \nabla u \nabla v dx + \int_{ \mathbb{R}^N} |u|^{p(x)-2} u v dx \\ &\quad - \lambda \int_{ \mathbb{R}^N} \varphi(x) |u|^{ \alpha(x) -2} uv dx - \int_{ \mathbb{R}^N} h v dx,\quad \forall u, v \in E. \end{align*} Hence, in order to obtain weak solutions of the problem \eqref{ePl} we will look for critical points of the functional $ J_{ \lambda}$. Now, we have to note that since $ \operatorname{meas}( \Omega_2) \neq 0$, then one cannot show that the functional $ J_{ \lambda} $ is coercive and consequently we cannot find a global minimum of the functional $ J_{ \lambda}$. The existence of a first critical point should be established using the Ekeland's variational principle. \begin{lemma} \label{lem3.1} Under the assumptions of Theorem \ref{thm1.1}, there exists $ \lambda_* > 0 $ such that for any $ 0 < \lambda < \lambda_*$, there exists $ \gamma_{ \lambda} > 0 $ and $ \eta_{ \lambda} > 0 $ such that $$ J_{\lambda}(u) \geq \gamma_{ \lambda}\ \text{for}\ \|u\| = \frac{1}{2}\quad \text{provided that } \|h\|_{-1} < \eta_{ \lambda}. $$ \end{lemma} \begin{proof} Let $ u \in W^{1, p( \cdot)}( \mathbb{R}^N) $ be such that $ \|u\| < 1$. By \eqref{e2.1}, \eqref{e2.2} and \eqref{e2.3} we have \begin{equation} \begin{aligned} \int_{ \Omega_1} \frac{ \varphi(x)}{ \alpha(x)}|u|^{ \alpha(x)} dx & \leq 2 | \varphi( \cdot)|_{L^{ \frac{p( \cdot)}{ p( \cdot) - \alpha( \cdot)}}( \Omega_1)} ||u|^{ \alpha(\cdot)}| _{L^{ \frac{ p( \cdot)}{ \alpha( \cdot)}}( \Omega_1)} \\ & \leq c_1 (|u|_{ L^{ p( \cdot)}( \Omega_1)}^{ \alpha_{ \Omega_1}^+} + |u|_{ L^{ p( \cdot)}( \Omega_1)}^{ \alpha_{ \Omega_1}^-}) \\ & \leq c_2 \|u\|_{ W^{1 , p( \cdot)}( \Omega_1)}^{ \alpha_{ \Omega_1}^-}, \end{aligned} \label{e3.1} \end{equation} and \begin{equation} \int_{ \Omega_2} \frac{ \varphi(x)}{ \alpha(x)}|u|^{ \alpha(x)} dx \leq 2 | \varphi( \cdot)|_{L^{ r( \cdot)}( \Omega_2)} | |u|^{ \alpha( \cdot)}|_{L^{ \frac{r( \cdot)}{ r( \cdot) -1}}( \Omega_2)} \leq c_3 \|u\|_{W^{1 , p ( \cdot)}( \Omega_2)}^{ \alpha_{ \Omega_2}^-}. \label{e3.2} \end{equation} Using again \eqref{e2.2} and \eqref{e2.3}, and taking \eqref{e3.1} and \eqref{e3.2} into account, we obtain \begin{equation} \begin{aligned} J_{ \lambda} (u) & \geq \frac{1}{p^+} (\|u\|_{W^{1,p( \cdot)} ( \Omega_1)}^{ p_ {\Omega_1}^+} + \|u\|_{W^{1,p( \cdot)} ( \Omega_2)}^{ p_ {\Omega_2}^+}) \\ &\quad - \lambda c_2 \|u\|_{ W^{1 , p( \cdot)} ( \Omega_1)}^{ \alpha_{ \Omega_1}^-} - \lambda c_3 \|u\|_{ W^{1 , p( \cdot)}( \Omega_2)}^{ \alpha_{ \Omega_2}^-} - \|h\|_{-1} \|u\| \\ & \geq \|u\|_{W^{1,p( \cdot)} ( \Omega_2)}^{ p_ {\Omega_2}^+} ( \frac{1}{p^+} - \lambda c_3 \|u\|_{ W^{1 , p( \cdot)}( \Omega_2)}^{ \alpha_{ \Omega_2}^- - p_{ \Omega_2}^+}) \\ &\quad + \frac{1}{p^+} \|u\|_{ W^{1 , p( \cdot)}( \Omega_1)}^{ p_{ \Omega_1}^+} - \lambda c_2 \|u\|_{ W^{1 , p( \cdot)}( \Omega_1)}^{ \alpha_{ \Omega_1}^-} - \|h\|_{-1} \|u\|. \end{aligned} \label{e3.3} \end{equation} For $ \lambda \leq \frac{1}{2 p^+ c_3}$, we have $$ \frac{1}{p^+} - \lambda c_3 \|u\|_{W^{1 , p( \cdot)} ( \Omega_2)}^{ \alpha_{ \Omega_2}^- - p_{ \Omega_2}^+} \geq \frac{1}{p^+} - \lambda c_3 \geq \frac{1}{2 p^+}. $$ Putting that inequality in \eqref{e3.3}, it yields \begin{equation} J_{ \lambda}(u) \geq c_4 \|u\|^{ \sup(p_ {\Omega_1}^+,\ p_{ \Omega_2}^+)} -c_2 \lambda \|u\|^{ \alpha_{ \Omega_1}^-} - \|h\|_{-1} \|u\|. \label{e3.4} \end{equation} Set \[ \lambda_* = \inf( \frac{1}{2p^+ c_3},\, \frac{c_4}{c_2} ( \frac{1}{2})^{ \sup(p_ {\Omega_1}^+,\ p_{ \Omega_2}^+) - \alpha_{ \Omega_1}^-}). \] For $ 0 < \lambda < \lambda_*$, set \begin{gather*} \gamma_{ \lambda} = c_4 ( \frac{1}{2})^{ \sup(p_ {\Omega_1}^+,\ p_{ \Omega_2}^+)} -c_2 \lambda ( \frac{1}{2})^{ \alpha_{ \Omega_1}^-} - \frac{\|h\|_{-1}}{2}, \\ \eta_{ \lambda} = 2 (c_4 ( \frac{1}{2})^{ \sup(p_ {\Omega_1}^+, p_{ \Omega_2}^+)} -c_2 \lambda ( \frac{1}{2})^{ \alpha_{ \Omega_1}^-}). \end{gather*} The claimed result can be deduced from \eqref{e3.4}. \end{proof} \begin{lemma} \label{lem3.2} Let $ (u_n)_n \subset W^{1, p( \cdot)}( \mathbb{R}^N) $ be a bounded sequence such that $ J'_{ \lambda}(u_n) \to 0$. Then$, (u_n)_n $ is relatively compact. \end{lemma} \begin{proof} Let $ u $ be the weak limit of $ (u_n)_n $ in $ W^{1, p( \cdot)}( \mathbb{R}^N)$. We claim that, up to a subsequence, $(u_n)_n $ is strongly convergent to $ u $ in $ W^{1, p( \cdot)}( \mathbb{R}^N)$. For $ t > 0$, denote $ B_t = \{x \in \mathbb{R}^N: |x| < t\}$. We have \begin{equation} \int_{ \Omega_2\backslash{B_t}} \varphi(x) | u_n -u|^{ \alpha(x)} dx \leq 2 ||u_n -u|^{ \alpha(\cdot)}|_{L^{ \frac{r( \cdot)}{r( \cdot)-1}} ( \mathbb{R}^N)} | \varphi( \cdot)|_{L^{r( \cdot)}( \Omega_2\backslash{B_t})}. \label{e3.5} \end{equation} Now, since $ \varphi \in L^{r( \cdot)}( \Omega_2)$, it follows that $ | \varphi( \cdot)|_{L^{r( \cdot)}( \Omega_2\backslash{B_t})} \to 0$ as $t \to + \infty$. Taking into account that $ (u_n)_n $ is bounded in $ W^{1, p( \cdot)}( \mathbb{R}^N)$, it follows from \eqref{e3.5} that for all $ \epsilon > 0 $ there exists $ t_{ \epsilon} > 0 $ large enough such that \begin{equation} \int_{ \Omega_2\backslash{B_{t_{ \epsilon}}}} \varphi(x) |u_n -u|^{ \alpha(x)} dx < \frac{ \epsilon}{2}. \label{e3.6} \end{equation} On the other hand, we have \begin{equation} \int_{ \Omega_2 \cap B_{t_{ \epsilon}}} \varphi(x) |u_n -u|^{ \alpha(x)} dx \leq \sup_{x \in B_{t_{ \epsilon}}} | \varphi(x)| \int_{ \Omega_2 \cap B_{t_{ \epsilon}}} |u_n -u|^{ \alpha(x)} dx. \label{e3.7} \end{equation} Since $ \alpha(x) < \frac{\alpha(x) r(x)}{r(x) -1} \leq p^*(x)$ for all $x \in \Omega_2 $ and $ ( \Omega_2 \cap B_{t_{ \epsilon}}) $ is a bounded open set of $ \Omega_2$, we obtain $$ \lim_{n \to + \infty} \int_{ \Omega_2 \cap B_{t_{\epsilon}}} |u_n -u|^{ \alpha(x)} dx = 0. $$ Having in mind that $ \varphi $ is continuous, then $ \sup_{x \in B_{t_{ \epsilon}}} | \varphi(x)| < + \infty$ and consequently we deduce from \eqref{e3.7} that $$ \lim_{n \to + \infty} \int_{ \Omega_2 \cap B_{t_{\epsilon}}} \varphi(x) |u_n -u|^{ \alpha(x)} dx = 0. $$ This implies that there exists $ n_0( \epsilon) \geq 1 $ such that for all $n \geq n_0( \epsilon)$, we have \begin{equation} \int_{ \Omega_2 \cap B_{t_{\epsilon}}} \varphi(x)|u_n -u|^{ \alpha(x)} dx < \frac{\epsilon}{2}. \label{e3.8} \end{equation} Combining \eqref{e3.6} and \eqref{e3.8}, it yields $$ \int_{ \Omega_2} \varphi(x) |u_n -u|^{ \alpha(x)} dx < \epsilon\quad \forall n \geq n_0( \epsilon). $$ Hence, \begin{equation} \lim_{n \to + \infty} \int_{ \Omega_2} \varphi(x) |u_n -u|^{ \alpha(x)} dx = 0. \label{e3.9} \end{equation} Next, if we replace $ r( \cdot) $ by $ \frac{ p(\cdot)}{ p( \cdot) - \alpha( \cdot)} $ and $ \frac{ r( \cdot)}{ r ( \cdot) -1} $ by $ p( \cdot)$, proceeding as previously (i.e. for the open set $ \Omega_2 $), we can so easily infer \begin{equation} \lim_{n \to + \infty} \int_{ \Omega_1} \varphi(x) |u_n-u|^{ \alpha(x)} dx = 0. \label{e3.10} \end{equation} On the other hand, since $ J'_{ \lambda}(u_n) \to 0$, we have \begin{equation} \begin{aligned} &\int_{ \mathbb{R}^N} | \nabla u_n|^{p(x) -2} \nabla u_n \nabla (u_n -u) dx + \int_{ \mathbb{R}^N} |u_n|^{p(x) -2} u_n (u_n -u) dx \\ & - \int_{ \mathbb{R}^N} \varphi(x) |u_n|^{ \alpha(x)-2} u_n ( u_n -u) dx - \int_{ \mathbb{R}^N} h(u_n -u) dx \to 0, \end{aligned} \label{e3.11} \end{equation} as $n \to + \infty$. Having in mind that $ u_n \rightharpoonup u $ weakly in $ W^{1, p( \cdot)}( \mathbb{R}^N)$, we deduce from \eqref{e3.11}, \eqref{e3.10} and \eqref{e3.9} that \begin{equation} \begin{aligned} 0 &\leq \int_{ \mathbb{R}^N} (| \nabla u_n|^{p(x)-2} \nabla u_n -| \nabla u|^{p(x)-2} \nabla u ) \nabla (u_n -u) dx \\ &\quad + \int_{ \mathbb{R}^N} (|u_n|^{p(x)-2} u_n - |u|^{p(x)-2} u) (u_n -u) dx \to 0,\quad \text{as } n \to + \infty. \end{aligned} \label{e3.12} \end{equation} Observe now that (see \cite{a1,f3,f5}), we have the following relations satisfied for $ \xi $ and $ \eta $ in $ \mathbb{R}^N $, \begin{equation} [( |\xi|^{p-2} \xi - | \eta |^{p-2} \eta )( \xi - \eta)]^{ \frac{p}{2}} (| \xi |^p + | \eta |^p )^{ \frac{2-p}{2}} \geq (p-1) | \xi - \eta|^p \label{e3.13} \end{equation} for $1 < p < 2 $ and \begin{equation} ( |\xi|^{p-2} \xi - | \eta |^{p-2} \eta )( \xi - \eta) \geq 2^{-p} | \xi - \eta |^p,\quad p \geq 2. \label{e3.14} \end{equation} Divide $ \mathbb{R}^N $ into two parts: $$ D_1 = \{x \in \mathbb{R}^N,\ p(x) < 2 \},\quad D_2 = \{x \in \mathbb{R}^N,\ p(x) \geq 2\}. $$ By \eqref{e3.12}, \eqref{e3.14} and \eqref{e2.4}, it yields \begin{equation} \lim_{n \to + \infty} \int_{D_2} (| \nabla u_n - \nabla u|^{p(x)} + |u_n -u|^{p(x)}) dx = 0. \label{e3.15} \end{equation} On the other hand, by \eqref{e3.13} we have \begin{align*} & \int_{D_1} | \nabla u_n - \nabla u|^{p(x)} dx \\ & \leq ( \frac{1}{p^--1}) \int_{D_1} (p(x)-1) | \nabla u_n - \nabla u|^{p(x)} dx \\ & \leq ( \frac{1}{p^--1}) \int_{D_1} ((| \nabla u_n|^{p(x)-2} \nabla u_n - | \nabla u|^{p(x)-2} \nabla u)(\nabla u_n - \nabla u))^{ \frac{p(x)}{2}} \\ & \quad \times (| \nabla u_n|^{p(x)} + | \nabla u|^{p(x)})^{ \frac{2-p(x)}{2}} dx. \end{align*} Using \eqref{e3.12} and \eqref{e2.4} and having in mind that $ (u_n)_n $ is bounded in $ E$, we deduce $$ \int_{D_1} | \nabla u_n - \nabla u|^{p(x)} dx \to 0,\quad \text{as } n \to + \infty. $$ Similarly, we obtain $$ \int_{D_1} |u_n-u|^{p(x)} dx \to 0, \quad \text{as}\ n \to + \infty. $$ Thus, \begin{equation} \int_{D_1} (| \nabla u_n - \nabla u|^{p(x)} + |u_n -u|^{p(x)}) dx \to 0,\quad \text{as } n \to + \infty. \label{e3.16} \end{equation} From \eqref{e3.15}, \eqref{e3.16} and \eqref{e2.4}, we conclude that $u_n \to u$ strongly in $W^{1, p( \cdot)}( \mathbb{R}^N)$. \end{proof} \subsection*{Completion of the proof of Theorem \ref{thm1.1}} Let $$ m_{ \lambda} = \inf\{ J_{ \lambda}(u),\ u \in W^{1, p( \cdot)} ( \mathbb{R}^N) \text{ and } \|u\| \leq \frac{1}{2} \}. $$ The set $$ \overline{B_{1/2}^{W^{1, p( \cdot)}( \mathbb{R}^N)}}(0) = \{u \in W^{1, p( \cdot)}( \mathbb{R}^N),\ \|u\| \leq \frac{1}{2} \} $$ is a complete metric space with respect to the distance $$ \operatorname{dist}(u,v) = \|u-v\|,\ u,\ v \in W^{1, p( \cdot)}( \mathbb{R}^N). $$ The functional $ J_{ \lambda} $ is lower semi-continuous and bounded from below in the set $ \overline{B_{1/2}^{W^{1, p( \cdot)}( \mathbb{R}^N)}}(0)$. Note, that $ \inf_{\|v\| < 1/2} J_{\lambda}(v) \leq J_{ \lambda}(0) = 0$ and $ \inf_{\|v\| = 1/2} J_{ \lambda} (v) \geq \gamma_{ \lambda} > 0$ (provided that $ \|h\|_{-1} < \eta_{ \lambda})$. Let \[ 0 < \epsilon < \inf_{\|v\| = 1/2} J_{ \lambda} (v) - \inf_{\|v\| < 1/2} J_{ \lambda} (v). \] Applying Ekeland's variational principle (see \cite{e3}), we can find $ u_ { \epsilon} \in \overline{B_{1/2}^{W^{1, p( \cdot)} ( \mathbb{R}^N)}}(0) $ such that $$ J_{ \lambda}(u_{ \epsilon}) < m_{ \lambda} + \epsilon,\quad J_{ \lambda}( u_ { \epsilon}) < J_{ \lambda}(w) + \epsilon \|w - u_{ \epsilon}\|, \quad\forall w \neq u_{ \epsilon}. $$ Since, $J_{ \lambda}( u_{ \epsilon}) \leq m_{ \lambda} + \epsilon \leq \inf_{\|v\| < 1/2} J_{ \lambda}(v) + \epsilon < \inf_{\|v\| = 1/2} J_{ \lambda} (v)$, it follows that $$ u_{ \epsilon} \in B_{1/2}^{W^{1, p( \cdot)}( \mathbb{R}^N)}(0) = \{u \in W^{1, p( \cdot)} (\mathbb{R}^N),\ \|u\|< \frac{1}{2}\}. $$ Define $ I_{ \lambda}^{ \epsilon} : \overline{B_{1/2} ^{W^{1, p( \cdot)}( \mathbb{R}^N)}}(0) \to \mathbb{R} $ by $ I_{ \lambda}^{ \epsilon}(u) = J_{ \lambda}(u) + \epsilon \|u - u_{ \epsilon}\|$. Obviously, $u_{ \epsilon} $ is a minimum of $ I_{ \lambda}^{ \epsilon}$. Then $$ \frac{I_{ \lambda}^{ \epsilon}(u_{ \epsilon} + t v) - I_{ \lambda}^{ \epsilon}(u_{ \epsilon})}{|t|} \geq 0,\quad \forall 0 < |t| < 1 \text{ and } v \in B_{1/2} ^{W^{1, p( \cdot)}( \mathbb{R}^N)}(0), $$ which implies $$ \frac{J_{ \lambda}(u_{ \epsilon} + t v) - J_{ \lambda}(u_{ \epsilon})}{|t|} + \epsilon \|v\| \geq 0. $$ Let $ t \to 0^+$, it follows that $ \langle J'_{ \lambda}(u_{ \epsilon}), v\rangle + \epsilon \|v\| \geq 0$. Next, let $ t \to 0^-; $ it follows that $ - \langle J'_{ \lambda}(u_{ \epsilon}), v\rangle + \epsilon \|v\| \geq 0$. Consequently, we obtain that $ \|J'_{ \lambda}( u_{ \epsilon})\| \leq \epsilon$. Hence, there exists a sequence $ (u_n)_n \subset B_{1/2}^{W^{1, p( \cdot)}( \mathbb{R}^N)}(0) $ such that $$ J_{ \lambda}(u_n) \to m_{ \lambda},\quad J'_{ \lambda}(u_n) \to 0. $$ Observing that $ (u_n)_n $ is bounded in $ W^{1, p( \cdot)}( \mathbb{R}^N) $ and using Lemma \ref{lem3.2}, we have that $ (u_n)_n $ is strongly convergent to its weak limit denoted, for example, by $ u_{0, \lambda} \in W^{1, p( \cdot)}( \mathbb{R}^N)$. Moreover, since $ J_{ \lambda} \in C^1( W^{1, p( \cdot)}( \mathbb{R}^N), \mathbb{R})$, it yields $ J_{ \lambda}(u_{0, \lambda}) = m_{ \lambda} $ and $ J'_{ \lambda}(u_{0, \lambda}) = 0$. Hence, $ u_{0, \lambda} $ is a weak solution of the problem \eqref{ePl}. Now, we claim that $ m_{ \lambda} < 0$. We distinguish two cases. * If (H3) holds. Let $ \psi $ be as in (H3). For $ 0 < t < 1$, we have $$ J_{ \lambda} (t \psi) \leq t^{\inf(p_{ \Omega_1}^-, p_{ \Omega_2}^-)} \int_{ \mathbb{R}^N} ( | \nabla \psi|^{p(x)} + | \psi|^{p(x)}) dx - t \int_{ \mathbb{R}^N} h(x) \psi(x) dx. $$ Since $ \inf(p_{ \Omega_1}^-, p_{ \Omega_2}^-) > 1$, we deduce that there exists $ 0 < t_0 < \inf(1,\frac{1}{2 \| \psi\|}) $ such that $ J_{ \lambda} (t_0 \psi) < 0$. Taking into account that $ t_0 \psi \in \overline{B_{1/2}^{W^{1, p( \cdot)}( \mathbb{R}^N)}}(0)$, it follows that $ m_{ \lambda} < 0$. * Assume that $ h = 0$. Let $ a_0 \in \Omega_1 $ and $ r_0 > 0 $ small enough be such that $ \overline{B_{r_0}(a_0)} \subset \Omega_1 $ and $ p_0 = \inf_{x \in \overline{B_{r_0}(a_0)}} p(x) > \alpha_0 = \sup_{x \in \overline{B_{r_0}(a_0)}} \alpha(x)$. Consider $ \xi \in C_0^{ \infty}( B_{r_0}(a_0)), \xi \neq 0$. For $ 0 < t < 1$, we have \begin{align*} J_{ \lambda} (t \xi) & \leq t^{ p_ {0}} \int_{ \Omega_1} \Big(| \nabla \xi|^{p(x)} + |\xi|^{p(x)}\Big) dx - \lambda t^{ \alpha_{ 0}} \int_{ \Omega_1} \frac{ \varphi(x)}{ \alpha(x)} |\xi|^{ \alpha(x)} dx \\ & \leq c_8 t^{ p_ {0}} -c_9 \lambda t^{ \alpha_{ 0}} \\ & \leq t^{ \alpha_{ 0}} ( c_8 t^{ p_ {0}- \alpha_{ 0}} - c_9 \lambda). \end{align*} Since$, p_{0}- \alpha_{0} > 0$, there exists $ 0 < t_1 ( \lambda) < \inf(1,\ \frac{1}{2 \| \xi\|}) $ such that $ J_{ \lambda}(t_1 ( \lambda) \xi) < 0$. Hence, $m_{ \lambda} \leq J_{ \lambda}(t_1( \lambda) \xi) < 0$. In this last case, by \eqref{e3.1} and \eqref{e3.2}, we have \begin{align*} \int_{ \mathbb{R}^N} (| \nabla u_{0, \lambda}|^{p(x)} + |u_{0, \lambda}|^{p(x)}) dx & = \lambda \Big(\int_{ \Omega_1} \varphi(x) |u_{0, \lambda}|^{ \alpha(x)} dx + \int_{ \Omega_2} \varphi(x) |u_{0, \lambda}|^{ \alpha(x)} dx\Big) \\ & \leq \lambda \Big(c_{10} \|u_{0, \lambda}\|_{W^{1, p( \cdot)} ( \Omega_1)}^{ \alpha_{ \Omega_1}^-} + c_{11} \|u_{0, \lambda}\|_{W^{1, p( \cdot)}( \Omega_2)}^{ \alpha_{ \Omega_2}^-}\Big) \\ & \leq \lambda \Big(c_{10} ( \frac{1}{2})^{ \alpha_{ \Omega_1}^-} + c_{11} ( \frac{1}{2})^{ \alpha_{ \Omega_2}^-}\Big). \end{align*} Using this inequality, it follows that $ \lim_{ \lambda \to 0} \|u_{0, \lambda}\| = 0$. This completes the proof of Theorem \ref{thm1.1}. \section{Proof of Theorem \ref{thm1.2}} Here, clearly $ E \neq W^{1, p( \cdot)}( \mathbb{R}^N)$. Moreover, the arguments used in the proof of Theorem \ref{thm1.1} are no longer valid. In fact, we cannot establish the existence of weak solution as a global neither a local minimum for the energy functional corresponding to the problem \eqref{ePl} and the Mountain-Pass is not useful as well. Hence, some new ideas have to be introduced and some new tools have to be employed. We shall adapt arguments used in \cite{z1}. \begin{lemma} \label{lem4.1} There is $ \lambda_{**} > 0 $ such that if $ 0 < \lambda < \lambda_{**}$, then there exists a nonnegative and nontrivial function $ \overline{U_{ \lambda}} \in E \cap L^{ \infty}( \mathbb{R}^N) $ satisfying $$ \int_{ \mathbb{R}^N} | \nabla \overline{U_{ \lambda}}|^{p(x)-2} \nabla \overline{U_{ \lambda}} \nabla w\,dx + \int_{ \mathbb{R}^N} (\overline{U_{ \lambda}})^{p(x)-1} w\,dx \geq \lambda \int_{ \mathbb{R}^N} \varphi(x) (\overline{U_{ \lambda}})^{ \alpha(x)-1} w\,dx, $$ for every $ w \in E $ with $ w \geq 0$. ($ \overline{U_{ \lambda}} $ is called a weak super-solution of \eqref{ePl}). \end{lemma} \begin{proof} For $ \lambda > 0$, define $ \overline{U_{ \lambda}}: \mathbb{R}^N \to \mathbb{R} $ by $$ \overline{U_{ \lambda}}(x) = \begin{cases} 1 & \text{if } |x| < 1 \\ 2 - |x| & \text{if } 1 \leq |x| \leq 2 \\ 0 & \text{if } |x| > 2. \end{cases} $$ For $ 1 \leq j \leq N$, we have $$ \frac{ \partial \overline{U_{ \lambda}}}{ \partial x_j}(x) = \begin{cases} 0 & \text{if } |x| < 1\ \text{or}\ |x| > 2 \\ - x_j/|x| & \text{if } 1 \leq |x| \leq 2, \end{cases} $$ where $ x = (x_1,\cdots,x_N)$. Thus, $$ | \nabla \overline{U_{ \lambda}}(x)| = \begin{cases} 0 & \text{if } |x| < 1 \text{ or } |x| > 2 \\ 1 & \text{if } 2 \leq |x| \leq 2. \end{cases} $$ Hence, \begin{align*} -\operatorname{div}(| \nabla \overline{U_{ \lambda}}|^{p(x)-2} \nabla \overline{U_{ \lambda}}) & = - \sum_{j=1}^N \frac{ \partial}{ \partial x_j} \Big(| \nabla \overline{U_{ \lambda}}|^{p(x)-2} \frac{ \partial \overline{U_{ \lambda}}}{ \partial x_j}\Big) \\ & = \begin{cases} 0 & \text{if } |x| < 1 \text{ or } |x| > 2 \\ \frac{N-1}{|x|} & \text{if } 1 \leq |x| \leq 2. \end{cases} \end{align*} Set $$ \lambda_{**} = \min\Big( \frac{1}{ \max_{|x| < 1} \varphi(x)}, \frac{N-1}{ \max_{1 \leq |x| \leq 2}( 2^{ \alpha(x)} \varphi(x))}\Big). $$ Then, for every $ 0 < \lambda < \lambda_{**}$, we have \begin{gather*} 1 \geq \lambda \varphi(x) \quad \text{if } |x| < 1 \\ \frac{N-1}{|x|} \geq \lambda \varphi(x) (2 - |x|)^{ \alpha(x)-1} \quad \text{if } 1 \leq |x| \leq 2. \end{gather*} Therefore, $$ -\operatorname{div}(| \nabla \overline{U_{ \lambda}}|^{p(x)-2} \nabla \overline{U_{ \lambda}}) + ( \overline{U_{ \lambda}})^{p(x)-1} \geq \lambda \varphi(x) (\overline{U_{ \lambda}})^{ \alpha(x)-1}. $$ This completes the proof. \end{proof} \subsection*{Completion of the proof of Theorem \ref{thm1.2}} For $ 0 < \lambda < \lambda_{**}$, set $$ f_{ \lambda}(x,s) = \lambda\ \varphi(x) |s|^{ \alpha(x)-2} s,\quad x \in \mathbb{R}^N,\; s \in \mathbb{R}. $$ Note that there exists $ L_{ \lambda} > 0 $ such that, for every $ s \in [-1, 1] $ and $ x \in \overline{B(0,2)} = \{x \in \mathbb{R}^N,\ |x| \leq 2\}$, we have $$ | \frac{ \partial f_{ \lambda}}{ \partial s}(x,s)| \leq L_{ \lambda}. $$ Thus, $(x,s) \longmapsto f_{ \lambda}(x,s) $ is $L_{ \lambda}-$Lipschitz continuous with respect to $ s \in [-1,1] $ uniformly for $ x \in \overline{B(0,2)}$; i.e., we have \begin{equation} f_{ \lambda}(x,s_1)-f_{ \lambda}(x,s_2) \leq L_{ \lambda} (s_2 -s_1), \label{e4.1} \end{equation} for any $ s_1, s_2 \in [-1,1] $ with $ s_1 \leq s_2 $ and $ x \in \overline{B(0,2)}$. Now, define $$ \tilde{f_{ \lambda}}(x,s) = \begin{cases} -f(x, \overline{U_{ \lambda}}(x))-L_{ \lambda} \overline{U_{ \lambda}}(x) & \text{if } s \leq - \overline{U_{ \lambda}}(x) \\ f_{ \lambda}(x,s) + L_{ \lambda} s & \text{if } -\overline{U_{ \lambda}}(x) < s \leq \overline{U_{ \lambda}}(x) \\ f_{ \lambda}(x, \overline{U_{ \lambda}}(x)) + L_{ \lambda} \overline{U_{ \lambda}}(x) & \text{if } s > \overline{U_{ \lambda}}(x), \end{cases} $$ and $ \tilde{F_{ \lambda}}(x, s) = \int_0^s \tilde{f_{ \lambda}}(x,t) dt$. If $ s \leq - \overline{U_{ \lambda}}(x)$, we have $$ \tilde{F_{ \lambda}}(x,s) \leq (-s) (f_{ \lambda} (x, \overline{U_{ \lambda}}(x)) + L_{ \lambda} \overline{U_{ \lambda}}(x)). $$ If $ 0 \leq s \leq \overline{U_{ \lambda}}(x)$, using \eqref{e4.1} and the fact that $ \|\overline{U_{ \lambda}}\|_{ \infty} = \sup_{x \in \mathbb{R}^N} |\overline{U_{ \lambda}}(x)| = 1$, we have $$ \tilde{F_{ \lambda}}(x,s) \leq (f_{ \lambda}(x,s) +L_{ \lambda} s) s \leq (f_{ \lambda}(x, \overline{U_{ \lambda}}(x)) + L_{ \lambda} \overline{U_{ \lambda}}(x)) s. $$ If $ - \overline{U_{ \lambda}}(x) < s < 0$, we have \begin{align*} \tilde{F_{ \lambda}}(x,s) \leq (f_{ \lambda}(x,s) +L_{ \lambda} s) s & \leq (f_{ \lambda}(x, -\overline{U_{ \lambda}}(x)) - L_{ \lambda} \overline{U_{ \lambda}}(x)) s \\ & \leq (f_{ \lambda}(x, \overline{U_{ \lambda}}(x)) + L_{ \lambda} \overline{U_{ \lambda}}(x)) (-s). \end{align*} If $ s > \overline{U_{ \lambda}}(x)$, we have \begin{align*} \tilde{F_{ \lambda}}(x,s) & = \int_0^{\overline{U_{ \lambda}}(x)} (f_{ \lambda}(x,t) + L_{ \lambda} t) dt + \int_{\overline{U_{ \lambda}}(x)}^s (f_{ \lambda}(x, \overline{U_{ \lambda}}(x)) + L_{ \lambda} \overline{U_{ \lambda}}(x))dt \\ & \leq (f_{ \lambda}(x, \overline{U_{ \lambda}}(x)) + L_{ \lambda} \overline{U_{ \lambda}}(x)) \overline{U_{ \lambda}}(x) + (f_{ \lambda}(x, \overline{U_{ \lambda}}(x)) + L_{ \lambda} \overline{U_{ \lambda}}(x)) (s- \overline{U_{ \lambda}}(x)) \\ & \leq (f_{ \lambda}(x, \overline{U_{ \lambda}}(x)) + L_{ \lambda} \overline{U_{ \lambda}}(x)) s. \end{align*} Therefore, for all $ (x,s) \in \mathbb{R}^N \times \mathbb{R}$, \begin{equation} \tilde{F_{ \lambda}}(x,s) \leq (f_{ \lambda}(x, \overline{U_{ \lambda}}(x)) + L_{ \lambda} \overline{U_{ \lambda}}(x)) |s|. \label{e4.2} \end{equation} Next, we introduce the functional space $ X = W^{1, p( \cdot)}( \mathbb{R}^N) \cap L^2( \mathbb{R}^N) $ equipped with the norm $$ \|u\|_X = \|u\|_{W^{1, p (\cdot)}( \mathbb{R}^N)} + |u|_{L^2( \mathbb{R}^N)}. $$ For any $ u \in X$, we define $$ \tilde{J_{ \lambda}}(u) = \int_{ \mathbb{R}^N} \frac{| \nabla u|^{p(x)} + |u|^{p(x)}}{p(x)} dx + \frac{L_{ \lambda}}{2} \int_{ \mathbb{R}^N} u^2 dx - \int_{ \mathbb{R}^N} \tilde{F_{ \lambda}}(x,u) dx. $$ Set $\psi_{ \lambda}(x) = (f_{ \lambda}(x, \overline{U_{ \lambda}}(x)) + L_{ \lambda} \overline{U_{ \lambda}}(x))$. Clearly, $\psi_{ \lambda} \in L^2( \mathbb{R}^N) $ and it becomes easy to verify that $ \tilde{J_{ \lambda}} \in C^1(X, \mathbb{R})$. By \eqref{e4.2}, for $ \epsilon > 0$, there exists $ c_{ \epsilon} > 0 $ such that $$ \tilde{J_{ \lambda}}(u) \geq \int_{ \mathbb{R}^N} \frac{| \nabla u|^{p(x)} + |u|^{p(x)}}{p(x)} dx + \frac{L_{ \lambda}}{2} \int_{ \mathbb{R}^N} u^2 dx - \epsilon \int_{ \mathbb{R}^N} u^2 dx -c_{ \epsilon} \int_{ \mathbb{R}^N} ( \psi_{ \lambda}(x))^2 dx. $$ Choosing $ \epsilon > 0 $ such that $ \frac{L_{ \lambda}}{2} - \epsilon > 0$, we infer that $ \tilde{J_{ \lambda}} $ is coercive. Let $ (u_n)_n $ be a minimizing sequence of $ \tilde{J_{ \lambda}}$, i.e. $(u_n)_n \subset X $ and $ \tilde{J_{ \lambda}}(u_n) \to \inf_{v \in X} \tilde{J_{ \lambda}}(v)> - \infty$. Since $ \tilde{J_{ \lambda}}$ is coercive, then $ (u_n)_n $ is bounded and there exists $ u \in E $ such that $ u_n\rightharpoonup u $ weakly in $ X$. By the mean value theorem, there exists some $ \theta_n $ between 0 and 1 such that \begin{equation} \begin{aligned} | \int_{ \mathbb{R}^N} ( \tilde{F_{ \lambda}}(x,u_n) - \tilde{F_{ \lambda}}(x,u)) dx| & = | \int_{ \mathbb{R}^N} \tilde{f_{ \lambda}}(x, \theta_n(u_n -u))(u_n-u) dx| \\ & \leq \int_{\mathbb{R}^N} \psi_{ \lambda}(x) |u_n -u| dx. \end{aligned} \label{e4.3} \end{equation} Let $ A $ be a measurable subset of $ \mathbb{R}^N$. Using H\"older's inequality we have $$ \int_A \psi_{ \lambda}(x) |u_n -u| dx \leq 2 | \psi_{ \lambda}( \cdot)|_{L^2( A)} |u_n -u|_{L^2( \mathbb{R}^N)}. $$ Since $(u_n -u)_n $ is bounded in $ L^2( \mathbb{R}^N) $ and $ \psi_{ \lambda} \in L^2( \mathbb{R}^N)$, it follows that the integral $ \int_A \psi_{ \lambda}(x) |u_n -u| dx$ is small uniformly in $ n $ when the measure of $ A $ is small. \\ On the other hand, for $ R > 0$, we have $$ \int_{ \mathbb{R}^N \backslash{B_R}} \psi_{ \lambda}(x) |u_n -u| dx \leq 2 |u_n -u|_{L^2( \mathbb{R}^N)} | \psi_{ \lambda} ( \cdot)|_{L^2( \mathbb{R}^N \backslash{B_R})}. $$ Since $ \psi_{ \lambda}( \cdot) \in L^2( \mathbb{R}^N)$, $$ \lim_{R \to + \infty} | \psi_{ \lambda}( \cdot)|_{L^2( \mathbb{R}^N \backslash{B_R})} =0. $$ This fact together with the boundedness of the sequence $ (|u_n -u|_{L^2( \mathbb{R}^N)})_n $ implies that for every $ \epsilon > 0$, there exists $ R_{ \epsilon} > 0 $ large enough such that $$ \int_{ \mathbb{R}^N \backslash{B_{R_{ \epsilon}}}} \psi_{ \lambda}(x) |u_n -u| dx < \epsilon. $$ Therefore, we get the equi-integrability of the sequence $ ( \psi_{ \lambda}( \cdot) |u_n -u|)_n$. By the virtue of Vitali's Theorem, we obtain $$ \lim_{n \to + \infty} \int_{ \mathbb{R}^N}\psi_{ \lambda}(x) |u_n -u| dx = 0. $$ By \eqref{e4.3}, we deduce that $$ \lim_{n \to + \infty} \int_{ \mathbb{R}^N} \tilde{F_{ \lambda}}(u_n) dx = \int_{ \mathbb{R}^N} \tilde{F_{ \lambda}}(u) dx. $$ This implies $$ \inf_{v \in X} \tilde{J_{ \lambda}}(v) \leq \tilde{J_{ \lambda}}(u) \leq \liminf_{n \to + \infty} \tilde{J_{ \lambda}}(u_n). $$ Consequently, $\tilde{J_{ \lambda}}(u) = \inf_{v \in X} \tilde{J_{ \lambda}}(v)$ and we have \begin{equation} \begin{aligned} \int_{ \mathbb{R}^N} | \nabla u|^{p(x)-2} \nabla u \nabla w\,dx & + \int_{ \mathbb{R}^N} |u|^{p(x)-2} u w\,dx + L_{ \lambda} \int_{ \mathbb{R}^N} u w\,dx \\ & = \int_{\mathbb{R}^N} \tilde{f_{ \lambda}}(x,u) w\,dx,\ \forall w \in X. \end{aligned} \label{e4.4} \end{equation} Now take $ w = (u -\overline{U_{ \lambda}})^+ = \max(u -\overline{U_{ \lambda}}, 0) $ in \eqref{e4.4}, and having in mind the definition of $ \overline{U_{ \lambda}}$, we get \begin{align*} & \int_{ \mathbb{R}^N} | \nabla \overline{U_{ \lambda}}|^{p(x)-2} \nabla \overline{U_{ \lambda}} \nabla (u -\overline{U_{ \lambda}})^+ dx + \int_{\mathbb{R}^N}( \overline{U_{ \lambda}})^{p(x)-1} (u -\overline{U_{ \lambda}})^+ dx \\ &\quad + L_{ \lambda} \int_{ \mathbb{R}^N} \overline{U_{ \lambda}} (u -\overline{U_{ \lambda}})^+ dx \\& \geq \int_{ \mathbb{R}^N} (f_{ \lambda}(x, \overline{U_{ \lambda}}) + L_{ \lambda} \overline{U_{ \lambda}})(u -\overline{U_{ \lambda}})^+ dx \\ & \geq \int_{ \mathbb{R}^N} \tilde{f_{ \lambda}}(x,u) (u -\overline{U_{ \lambda}})^+ dx \\ & \geq \int_{ \mathbb{R}^N} | \nabla u|^{p(x)-2} \nabla u \nabla (u -\overline{U_{ \lambda}})^+ dx + \int_{ \mathbb{R}^N} |u|^{p(x)-2} u (u -\overline{U_{ \lambda}})^+ dx \\ &\quad + L_{ \lambda} \int_{ \mathbb{R}^N} u (u -\overline{U_{ \lambda}})^+ dx. \end{align*} Thus, \begin{align*} & \int_{ \mathbb{R}^N} (| \nabla u|^{p(x)-2} \nabla u - | \nabla \overline{U_{ \lambda}}|^{p(x)-2} \nabla \overline{U_{ \lambda}}) \nabla (u -\overline{U_{ \lambda}})^+ dx \\ & + \int_{ \mathbb{R}^N} (|u|^{p(x)-2} u - | \overline{U_{ \lambda}}|^{p(x)-2} \overline{U_{ \lambda}})(u -\overline{U_{ \lambda}})^+dx \\ & + L_{ \lambda} \int_{ \mathbb{R}^N} ((u- \overline{U_{ \lambda}})^+)^2 dx \leq 0. \end{align*} Taking into account that the terms $$ \int_{ \mathbb{R}^N} (| \nabla u|^{p(x)-2} \nabla u - | \nabla \overline{U_{ \lambda}}|^{p(x)-2} \nabla \overline{U_{ \lambda}}) \nabla (u -\overline{U_{ \lambda}})^+ dx $$ and $$ \int_{ \mathbb{R}^N} (|u|^{p(x)-2} u - | \overline{U_{ \lambda}}|^{p(x)-2} \overline{U_{ \lambda}})(u -\overline{U_{ \lambda}})^+dx $$ are nonnegative, then $ u \leq \overline{U_{ \lambda}} $ a.e. in $ \mathbb{R}^N$. On the other hand, define $ - \overline{U_{ \lambda}} = \overline{V_{ \lambda}}$, and take $ w = ( \overline{V_{ \lambda}}-u)^+ = \max( \overline{V_{ \lambda}}-u, 0) $ in \eqref{e4.4}, we obtain \begin{align*} &\int_{ \mathbb{R}^N} | \nabla \overline{V_{ \lambda}}|^{p(x)-2} \nabla \overline{V_{ \lambda}} \nabla ( \overline{V_{ \lambda}} -u)^+ dx + \int_{ \mathbb{R}^N} | \overline{V_{ \lambda}}|^{p(x)-2} \overline{V_{ \lambda}} (\overline{V_{ \lambda}} -u)^+ dx \\ & + L_{ \lambda} \int_{ \mathbb{R}^N} \overline{V_{ \lambda}} (\overline{V_{ \lambda}} -u)^+ dx \\ & \leq \int_{ \mathbb{R}^N} (f_{ \lambda}(x, \overline{V_{ \lambda}}) + L_{ \lambda} \overline{V_{ \lambda}}) (\overline{V_{ \lambda}} -u)^+ dx \\ & \leq \int_{ \mathbb{R}^N} \tilde{f_{ \lambda}}(x, u) (\overline{V_{ \lambda}} -u)^+ dx \\ & \leq \int_{ \mathbb{R}^N} | \nabla u|^{p(x)-2} \nabla u \nabla (\overline{V_{ \lambda}} -u)^+ dx + \int_{ \mathbb{R}^N} |u|^{p(x)-2} u (\overline{V_{ \lambda}} -u)^+ dx\\ &\quad + L_{ \lambda} \int_{ \mathbb{R}^N} u (\overline{V_{ \lambda}} -u)^+ dx. \end{align*} Thus, \begin{align*} & \int_{ \mathbb{R}^N} (| \nabla \overline{V_{ \lambda}}|^{p(x)-2} \nabla \overline{V_{ \lambda}} - | \nabla u|^{p(x)-2} \nabla u ) \nabla (\overline{V_{ \lambda}} - u )^+ dx \\ & + \int_{ \mathbb{R}^N} (| \overline{V_{ \lambda}}|^{p(x)-2} \overline{V_{ \lambda}} - |u|^{p(x)-2} u )(\overline{V_{ \lambda}}-u)^+dx \\ & + L_{ \lambda} \int_{ \mathbb{R}^N} (( \overline{V_{ \lambda}} -u)^+)^2 dx \leq 0. \end{align*} Hence, $( \overline{V_{ \lambda}} -u)^+ = 0$, which implies $ -\overline{U_{ \lambda}} \leq u $ a.e. in $ \mathbb{R}^N$. Therefore, $\tilde{ f_{ \lambda}}(x,u) = f_{ \lambda}(x, u) + L_{ \lambda} u $ and by \eqref{e4.4}, for all $ w \in X $ we have $$ \int_{ \mathbb{R}^N} | \nabla u|^{p(x)-2} \nabla u \nabla w\,dx + \int_{ \mathbb{R}^N} |u|^{p(x)-2} u w\,dx = \int_{ \mathbb{R}^N} f_{ \lambda}(x,u) w\,dx. $$ Now, without loss of generality, we could assume that $ 0 \in \Omega_1$. Taking into account that $ \Omega_1 $ is an open set, one can find $ 0 < r < 1 $ small enough such that $ \overline{B_r(0)} \subset \Omega_1 $ and $ p_1 = \inf_{x \in \overline{B_r(0)}} p(x) > \alpha_1 = \sup_{x \in \overline{ B_r(0)}} \alpha(x) $. Let $ \vartheta \in C_0^{ \infty}(B_r(0)) $ be such that $ \vartheta \neq 0 $ and $ \vartheta \geq 0$. Take $ 0 < t < 1 $ such that $ t \vartheta(x) \leq 1$, for all $x \in B_r(0)$. We have $\tilde{F_{ \lambda}}(x, t \vartheta(x)) = \int_0^{t \vartheta(x)} \tilde{f_{ \lambda}}(x,s) ds$. For $ x \notin B_r(0)$, $\tilde{F_{ \lambda}}(x, t \vartheta(x)) = 0$. For $ x \in B_r(0)$, $0 \leq t \vartheta(x) \leq \overline{U_{ \lambda}}(x) $ and $ \tilde{F_{ \lambda}}(x, t \vartheta(x)) = \lambda \frac{ \varphi(x)}{ \alpha(x)} t^{ \alpha(x)} |\vartheta(x)|^{ \alpha(x)} + \frac{L_{ \lambda}}{2} t^2 (\vartheta(x))^2$. Thus, we have \begin{align*} \tilde{J_{ \lambda}}(t \vartheta) & \leq t^{p_1} \int_{B_r(0)} (| \nabla \vartheta|^{p(x)} + |\vartheta|^{p(x)}) dx - \lambda t^{ \alpha_1} \int_{B_r(0)} \frac{ \varphi(x)}{ \alpha(x)} |\vartheta|^{ \alpha(x)} dx \\ & \leq t^{\alpha_1}( c_{12} t^{ p_1 - \alpha_1} - \lambda c_{13} ). \end{align*} Since $\ p_1 - \alpha_1 > 0$, then there exists $ 0 < t( \lambda) < 1 $ small enough such that $ \tilde{J_{ \lambda}}(t( \lambda) \vartheta) < 0$. Therefore, $\tilde{J_{ \lambda}} (u) = \inf_{v \in X} \tilde{J_{ \lambda}}(v) < 0$ and $ u \neq 0$. Now, note that $ u $ satisfies $$ \int_{ \mathbb{R}^N} | \nabla u|^{p(x)-2} \nabla u \nabla w\,dx + \int_{ \mathbb{R}^N} |u|^{p(x)-2} u w\,dx = \int_{ \mathbb{R}^N} f_{ \lambda}(x,u) w\,dx, $$ for all $w \in C_0^{ \infty}( \mathbb{R}^N)$. On the other hand, since $ |u| \leq \overline{U_{ \lambda}}$, then $ u \in E$. Having in mind that $ p(\cdot) $ satisfies the logarithmic H\"older inequality, we could immediately deduce that $ C_0^{ \infty}( \mathbb{R}^N) $ is dense in $ E $ and we infer $$ \int_{ \mathbb{R}^N} | \nabla u|^{p(x)-2} \nabla u \nabla w\,dx + \int_{ \mathbb{R}^N} |u|^{p(x)-2} u w\,dx = \lambda \int_{ \mathbb{R}^N} \varphi(x) |u|^{p(x)-2} u w \,dx,\quad $$ for all $w \in E$. This competes the proof of Theorem \ref{thm1.2}. \begin{thebibliography}{99} \bibitem{a1} S. Antontsev, S. Shmarev; \emph{Handbook of Differential Equations}, Stationary Partial Differential Equations, Volume III, Chap. 1, 2006. \bibitem{d1} L. Diening, P. Harjulehto, P. H\"ast\"o, M. 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The choice of the function $$ \overline{U_{ \lambda}}(x) = \begin{cases} 1 & \text{if } |x| < 1 \\ 2 - |x| & \text{if } 1 \leq |x| \leq 2 \\ 0 & \text{if } |x| > 2 \end{cases} $$ as a super-solution of the problem \eqref{ePl} is not appropriate since the identity $$ -\operatorname{div}\big(| \nabla \overline{U_{ \lambda}}|^{p(x)-2} \nabla \overline{U_{ \lambda}}\big) = \begin{cases} 0 & \text{if } |x| < 1 \text{ or } |x| > 2 \\ \frac{N-1}{|x|} & \text{if } 1 \leq |x| \leq 2 \end{cases} $$ is wrong. Some Dirac measures appear when computing $ -\operatorname{div}\big(| \nabla \overline{U_{ \lambda}}|^{p(x)-2} \nabla \overline{U_{ \lambda}}\big) $, in the sense of distributions. Thus, we have to change the choice of this function. For this purpose, we add the following assumption to Theorem \ref{thm1.2}, \begin{itemize} \item[(H6)] There exists a nonnegative and nontrivial function $ e$ in the space $L^{ \infty} ( \mathbb{R}^N) \cap W^{-1, p' (\cdot)}( \mathbb{R}^N) $ (where $ W^{-1, p'( \cdot)}( \mathbb{R}^N) $ is the dual space of $ W^{1, p( \cdot)}( \mathbb{R}^N)) $ such that $$ e(x) \geq \varphi(x),\quad \forall x \in \mathbb{R}^N. $$ \end{itemize} Concerning the construction of a super-solution of problem \eqref{ePl}, we note that the problem $$ -\operatorname{div}\big(| \nabla u|^{p(x)-2} \nabla u\big) + |u|^{p(x)-2} u = e $$ has a nontrivial and nonnegative weak solution $ U_e \in W^{1, p( \cdot)}( \mathbb{R}^N)$; i.e., $ U_e $ satisfies $$ \int_{ \mathbb{R}^N} | \nabla U_e|^{p(x)-2} \nabla U_e \nabla w dx + \int_{ \mathbb{R}^N} \left(U_e\right)^{p(x)-1} w dx = \int_{ \mathbb{R}^N} e(x) w(x) dx, $$ for all $w \in W^{1, p( \cdot)}( \mathbb{R}^N)$. Moreover, it is easy to see that $ U_e \in L^{ \infty}( \mathbb{R}^N) $ and that $ U_e \in E$. Let $$ \lambda_{**} = \frac{1}{\|U_e\|_{ \infty}^{ \alpha^+-1} + \|U_e\|_{ \infty}^{ \alpha^--1}}. $$ If $ 0 < \lambda < \lambda_{**}$, we have $ e(x) \geq \varphi(x) \geq \lambda \varphi(x) \left(U_e\right)^{ \alpha(x)-1}$. By the definition of $ U_e$, it follows immediately that $ U_e $ is a super-solution of the problem \eqref{ePl} provided that $ h= 0 $ and $ 0 < \lambda < \lambda_{**}$. Therefore, in the proof of Theorem \ref{thm1.2} we can take $ \overline{U_{ \lambda}} = U_e$, for all $0 < \lambda < \lambda_{**}$. Consequently, we can easily find a constant $ L_{ \lambda} $ such that $ f_{ \lambda}(x,s) $ is $ L_{\lambda}$-Lipschitz continuous with respect to $ s \in [- \|U_e\|_{ \infty}, \|U_e\|_{ \infty}] $ uniformly for $ x \in \mathbb{R}^N$. \medskip End of corrigendum. \end{document}