Semilinear elliptic equations with singular nonlinearities

We prove existence, regularity and nonexistence results for problems whose model is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\Delta u = \frac{f(x)}{u^{\gamma}}\quad {{\rm in}\,\Omega},$$\end{document}with zero Dirichlet conditions on the boundary of an open, bounded subset Ω of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}^{N}}$$\end{document}. 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).