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

Abstract

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

\[ \left\{ \begin{alignedat}{2} -\Delta _{p}u&= g(x,u)-\frac{h(x)}{u^{\alpha }}&\quad &\text{in }\Omega \\ u&= 0&&\text{on }\partial \Omega ,% \end{alignedat}% \right. \]

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