Asymptotic behavior and existence of solutions for singular elliptic equations

We study the asymptotic behavior, as γ tends to infinity, of solutions for the homogeneous Dirichlet problem associated with singular semilinear elliptic equations whose model is -Δu=f(x)uγinΩ,where Ω 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 Ω or only nonnegative. Through this study, we deduce optimal existence results of positive solutions for the homogeneous Dirichlet problem associated with -Δv+|∇v|2v=finΩ