Existence and multiplicity results for nonlinear problems involving the p(x)-Laplace operator

Abstract

In this paper we study the following nonlinear boundary-value problem $-\Delta_{p(x)} u=\lambda f(x,u) \quad \text{ in } \Omega,$ $|\nabla u|^{p(x)-2}\frac{\partial u}{\partial \nu}+\beta(x)|u|^{p(x)-2}u=\mu g(x,u) \quad \text{ on } \partial\Omega,$ where $\Omega\subset\mathbb{R}^N$ is a bounded domain with smooth boundary $\partial\Omega$, $\frac{\partial u}{\partial\nu}$ is the outer unit normal derivative on $\partial\Omega$, $\lambda, \mu$ are two real numbers such that $\lambda^{2}+\mu^{2}\neq0$, $p$ is a continuous function on $\overline{\Omega}$ with $\inf_{x\in \overline{\Omega}} p(x)\gt 1$, $\beta\in L^{\infty}(\partial\Omega)$ with $\beta^{-}:=\inf_{x\in \partial\Omega}\beta(x)\gt 0$ and $f : \Omega\times\mathbb{R}\rightarrow \mathbb{R}$, $g : \partial\Omega\times\mathbb{R}\rightarrow \mathbb{R}$ are continuous functions. Under appropriate assumptions on $f$ and $g$, we obtain the existence and multiplicity of solutions using the variational method. The positive solution of the problem is also considered.