Existence of solutions for a Steklov proble involving the $p(x)$-Laplacian

Abstract

By applying two versions of Mountain Pass Theorem, we prove two different situations of the existence of solutions for the following Steklov problem $\Delta_{p(x)}u =|u|^{p(x)-2}u$ in $\Omega$, $|\nabla u|^{p(x)-2}\frac{\partial u}{\partial \nu}= \lambda |u|^{q(x)-2}u$ on $\partial\Omega$, where $\Omega$ is a bounded domain in $\mathbb{R}^{N}(N\geq 2)$ with smooth boundary $\partial\Omega$ and $p(.), q(.):\bar{\Omega}\rightarrow (1, +\infty)$ are continuous functions.