Semilinear elliptic equations with singular nonlinearities

We prove existence, regularity and nonexistence results for problems whose model is $$-Delta u = frac{f(x)}{u^{gamma}}quad {{rm in},Omega},$$ with zero Dirichlet conditions on the boundary of an open, bounded subset Ω of ${mathbb{R}^{N}}$ . Here γ > 0 and f is a nonnegative function on Ω. Our results will depend on the summability of f in some Lebesgue spaces, and on the values of γ (which can be equal, larger or smaller than 1).