Asymptotic behavior and existence of solutions 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 $$-\mathrm{\Delta }u=\frac{f(x)}{u^{\gamma }}\text{ in }\mathrm{\Omega },$$ where $\mathrm{\Omega }$ is an open, bounded subset of $\text{RN}$ 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 $\mathrm{\Omega }$ or only nonnegative. Through this study we deduce optimal existence results of positive solutions for the homogeneous Dirichlet problem associated to $$-\mathrm{\Delta }v+\frac{\mathrm{\mid }\mathrm{\nabla }v{\mathrm{\mid }}^2}{v}=f\text{ in }\mathrm{\Omega }\mathrm{.}$$