Asymptotic behavior and existence of solutions for singular elliptic equations

Abstract

We study the asymptotic behavior, as $$\gamma $$ tends to infinity, of solutions for the homogeneous Dirichlet problem associated with singular semilinear elliptic equations whose model is $$\begin{aligned} -\Delta u=\frac{f(x)}{u^\gamma }\,\text { in }\Omega , \end{aligned}$$ where $$\Omega $$ is an open, bounded subset of $${\mathbb {R}}^{N}$$ and f is a bounded function. We deal with the existence of a limit equation under two different assumptions on f: either strictly positive on every compactly contained subset of $$\Omega $$ or only nonnegative. Through this study, we deduce optimal existence results of positive solutions for the homogeneous Dirichlet problem associated with $$\begin{aligned} -\Delta v + \frac{|\nabla v|^2}{v} = f\,\text { in }\Omega . \end{aligned}$$