On a singular logistic equation with the \(p\)-Laplacian (Q360708)

Summary: We prove the existence and nonexistence of positive solutions for the boundary value problems \[ \begin{cases} -\Delta_pu=g(x,u)-\frac{h(x)}{u^\alpha}&\quad\text{in }\Omega\\\quad\quad u=0&\quad\text{on }\partial\Omega\end{cases} \] where \(\Delta_pu=\mathrm{div}(| \nabla u|^{p-2}\nabla u),p>1\), \(\Omega\) is a bounded domain in \(\mathbb R^n\) with smooth boundary \(\partial\Omega\), \(\alpha\in(0,1),g:\Omega\times(0,\infty)\to\mathbb R\) is possibly singular at \(u=0\). An application to a singular logistic-like equation is given.