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

In this paper, we study the existence of positive solutions of the following equation\begin{document}$\begin{equation} (P_{\lambda}) \left\{\begin{array}{rclll}- \Delta_{p(x)} u+V(x)\vert u\vert^{p(x)-2}u & = & \lambda k(x) \vert u\vert^{\alpha(x)-2}u\\ &+& h(x) \vert u\vert^{\beta(x)-2}u&\mbox{ in }&\Omega\\u& = &0 &\mbox{ on }& \partial \Omega.\end{array}\right.\;\;\;\;\;\;\;\;\;\;\;\;\;\left( 1 \right) \end{equation}$\end{document}The study of the problem \begin{document}$ (P_{\lambda}) $\end{document} needs generalized Lebesgue and Sobolev spaces. In this work, under suitable assumptions, we prove that some variational methods still work. We use them to prove the existence of positive solutions to the problem \begin{document}$ (P_{\lambda}) $\end{document} in \begin{document}$ W_{0}^{1,p(x)}(\Omega) $\end{document} .