Existence of entropy solutions for anisotropic quasilinear degenerated elliptic problems with Hardy potential

Abstract

The aim of this work is to give an existence result of entropy solutions for anisotropic quasilinear degenerated elliptic problems of the form $$\begin{aligned} -\text{ div } (a(x,u,\nabla u)) + |u|^{s-1}u= f +\rho \frac{|u|^{p_{0}-2}u}{|x|^{p_{0}}}, \quad \text{ in } \ \ \varOmega , \end{aligned}$$ where $$-\text{ div } (a(x,u,\nabla u))$$ is a Leray-Lions operator from $$W_{0}^{1,(p_{i})}(\varOmega ,w)$$ to its dual, $$\varOmega $$ is an open bounded subset of $$I\!\!R^{N}$$ $$(N\ge 2)$$ containing the origin, the datum f is assumed to be merely integrable and $$\rho $$ is a positive constant.