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, $-{△}_{p}u=\lambda |u{|}^{p-2}u-\gamma u^{-\alpha }$ 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 $\lambda \le 0,\gamma <0$ and $\Omega$ is a bounded domain, with smooth boundary.