On the existence solutions for some Nonlinear elliptic problem

Abstract

In the present paper, we study the existence and regularity of positive solutions for the following boundary value problem : $\mathrm{-div}\> \big( \lvert\nabla u\rvert^{p-2}\nabla u ) + u^{s} = \dfrac{f}{u^{\alpha}}\mbox{ in } \Omega \mbox{ and } u=0\mbox{ on } \partial\Omega,$ where $ \Omega $ is an open and bounded subset of $ \mathbb{R}^{N} $ $ (N> p>1) $, $ 0<\alpha\leq 1 $, $ s\geq 1 $ and $f$ is a nonnegative function that belongs to some Lebesgue space.