Existence and Multiplicity of Positive Solutions for Singular <em>p</em>-Laplacian Equations

Abstract

Positive solutions are obtained for the boundary value problem \begin{alignat*}{2} \left\{ \begin{aligned} -\Delta _{p}u&= \lambda (u^{\beta }+\tfrac{1}{u^{\alpha }}) &\; &\hbox{in } \Omega \\ u&> 0 &\; &\hbox{in } \Omega \\ u&= 0 &\; &\hbox{on }\partial \Omega\,, \end{aligned}% \right. \tag{$*$} \end{alignat*} where $\Delta _{p}u=\dive(| \nabla u| ^{p-2}\nabla u)$, $1<p<N$, $N\geq 3$, $\Omega \subset {\mathbb R}^{N}$ is a bounded domain, $0<\alpha <1$ and $p-1<\beta <p^{\ast }-1$ \big($p^{\ast }=\frac{Np}{N-p}$\big) are two constants, $\lambda >0$ is a real parameter. We obtain that Problem ($*$) has two positive weakly solutions if $\lambda$ is small enough.