On a Singular Logistic Equation with the $p$-Laplacian

Abstract

We prove the existence and nonexistence of positive solutions for the boundary value problems

where ${\Delta }_{p}u=\mathrm{div}(|\nabla u{|}^{p-2}\nabla u),p>1,\Omega$ is a bounded domain in ${\mathbb{R}}^n$ with smooth boundary $\partial \Omega$ , $\alpha \in (0,1)$, $g:\Omega \times (0,\infty )\to \mathbb{R}$ is possibly singular at $u=0$. An application to a singular logistic-like equation is given.