Existence and Nonexistence results for singular elliptic equations

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 $\mathbb{R}^N$ and $f$ is a bounded function. We prove existence and nonexistence 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 and nonexistence results of positive solutions for the homogeneous Dirichlet problem associated to $$ -\Delta v + \frac{|\nabla v|^2}{v} = f\,\text{ in }\Omega\,. $$