Existence and Multiplicity of Positive Solutions for Singular $p$-Laplacian Equations

Abstract

Positive solutions are obtained for the boundary value problem

where ${\Delta }_{p}u=\mathrm{div}(|\nabla u{|}^{p-2}\nabla u)$, $1<p<N$, $N\ge 3$, $\Omega \subset {\mathbb{R}}^N$ is a bounded domain, $0<\alpha <1$ and $p-1<\beta <p^{\ast }-1$ ($p^{\ast }=\frac{Np}{N-p}$) are two constants, $\lambda >0$ is a real parameter. We obtain that Problem ($\ast$) has two positive weakly solutions if $\lambda$ is small enough.