On a Singular Logistic Equation with the $p$-Laplacian
On a Singular Logistic Equation with the $p$-Laplacian
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=\mathrm{div}(|\nabla u|^{p-2}\nabla u),p>1,\ \Omega is a bounded domain in \mathbb{R}^{n} with smooth boundary \partial \Omega , \alpha \in (0,1) …