Existence and nonexistence of solutions for a singular $p$-Laplacian Dirichlet problem

Abstract

We study the existence of positive radially symmetric solution for the singular $p$-Laplacian Dirichlet problem, $-\bigtriangleup_p u =\lambda |u|^{p-2} u-\gamma u^{-\alpha}$ where $\lambda>0,\gamma>0$ and, $0<\alpha<1$, are parameters and $\Omega$, the domain of the equation, is a ball in $\mathbb{R}^N$. By using some variational methods we show that, if $\lambda$ is contained in some interval, then the problem has a radially symmetric positive solution on the ball. Moreover, we obtain a nonexistence result, whenever $\lambda \leq 0, \gamma<0$ and $\Omega$ is a bounded domain, with smooth boundary.