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 to singular semilinear elliptic equations whose model is $$-\Delta u=\frac{f(x)}{u^\gamma}\,\text{ in }\Omega,$$ where $\Omega$ is an open, bounded subset of \RN 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 to $$-\Delta v + \frac{|\nabla v|^2}{v} = f\,\text{ in }\Omega.$$Comment: 31 page