Positive solutions for the $ p(x)- $Laplacian : Application of the Nehari method

Abstract

<p style='text-indent:20px;'>In this paper, we study the existence of positive solutions of the following equation</p><p style='text-indent:20px;'><disp-formula><label/><tex-math id="FE10000">\begin{document}$\begin{equation} (P_{\lambda}) \left\{ \begin{array}{rclll} - \Delta_{p(x)} u+V(x)\vert u\vert^{p(x)-2}u &amp; = &amp; \lambda k(x) \vert u\vert^{\alpha(x)-2}u\\ &amp;+&amp; h(x) \vert u\vert^{\beta(x)-2}u&amp;\mbox{ in }&amp;\Omega\\ u&amp; = &amp;0 &amp;\mbox{ on }&amp; \partial \Omega. \end{array} \right.\;\;\;\;\;\;\;\;\;\;\;\;\;\left( 1 \right) \end{equation}$\end{document}</tex-math></disp-formula></p> <p style='text-indent:20px;'>The study of the problem <inline-formula><tex-math id="M2">\begin{document}$ (P_{\lambda}) $\end{document}</tex-math></inline-formula> needs generalized Lebesgue and Sobolev spaces. In this work, under suitable assumptions, we prove that some variational methods still work. We use them to prove the existence of positive solutions to the problem <inline-formula><tex-math id="M3">\begin{document}$ (P_{\lambda}) $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M4">\begin{document}$ W_{0}^{1,p(x)}(\Omega) $\end{document}</tex-math></inline-formula>.</p>