Existence of solutions in the sense of distributions of anisotropic nonlinear elliptic equations with variable exponent

Abstract

The aim of this paper is to study the existence of solutions in the sense of distributions for a strongly nonlinear elliptic problem where the second term of the equation $f$ is in $ W^{-1,\overrightarrow{p}'(\,\cdot\,)}(\Omega)$ which is the dual space of the anisotropic Sobolev $W_{0}^{1,\overrightarrow{p}(\,\cdot\,)}(\Omega)$ and later $f$ will be in $L^{1}(\Omega)$.