Well-Posedness Result For a Nonlinear Elliptic Problem Involving Variable Exponent and Robin Type Boundary Condition

Abstract

In this work we study the following nonlinear elliptic boundary value problem, $b(u)-div \; a(x,\nabla u)=f \hbox{ in }\Omega$, $a(x,\nabla u).\eta=-\left|u\right|^{p(x)-2}u \hbox{ on }\partial \Omega$, where $\Omega$ is a smooth bounded open domain in $\mathbb{R}^{N}$, $N \geq 1$ with smooth boundary $\partial\Omega$. We prove the existence and uniqueness of a weak solution for $f \in L^{\infty}(\Omega)$, the existence and uniqueness of an entropy solution for $L^{1}$-data $f$. The functional setting involves Lebesgue and Sobolev spaces with variable exponent.