Singular elliptic problem involving a Hardy potential and lower order term

Abstract

We consider the following non-linear singular elliptic problem $$\begin{array}{}\text{(1)}& \left\{\begin{array}{cc}-div\mathrm{(}\mathit{M}\mathrm{(}\mathit{x}\mathrm{)}|\nabla \mathit{u}{|}^{\mathit{p}-2}\nabla \mathit{u}\mathrm{)}+\mathit{b}|\mathit{u}{|}^{\mathit{r}-2}\mathit{u}=\mathit{a}\frac{{\mathit{u}}^{\mathit{p}-1}}{|\mathit{x}{|}^{\mathit{p}}}+\frac{\mathit{f}}{{\mathit{u}}^{\mathit{\gamma }}}\hfill & \text{in }\mathrm{\Omega }\hfill \\ \mathit{u}\mathrm{>}0\hfill & \text{in }\mathrm{\Omega }\hfill \\ \mathit{u}=0\hfill & \text{on }\partial \mathrm{\Omega }\mathrm{,}\hfill \end{array}\right.\end{array}$$ where $1\mathrm{<}\mathit{p}\mathrm{<}\mathit{N}$ ; $\mathrm{\Omega }\subset {\mathbb{R}}^{\mathit{N}}$ is a bounded regular domain containing the origin and $0\mathrm{<}\mathit{\gamma }\mathrm{<}1$ , $\mathit{a}⩾0\mathrm{,}\mathit{b}\mathrm{>}0\mathrm{,}0⩽\mathit{f}\in {\mathit{L}}^{\mathit{m}}\mathrm{(}\mathrm{\Omega }\mathrm{)}\text{ and }1\mathrm{<}\mathit{m}\mathrm{<}\frac{\mathit{N}}{\mathit{p}}$ . The main goal of this work is to study the existence and regularizing effect of some lower order terms in Dirichlet problems despite the presence of Hardy the potentials and the singular term in the right hand side.