Nonlinear elliptic equations with singular nonlinearities

In this paper we study nonlinear elliptic boundary value problems with singular nonlinearities whose simplest example is [GRAPHICS] where Omega is a bounded open set in R-N (N >= 2),gamma > 0, 1 < p < N, 0 <= f is an element of L-m(Omega), m >= 1. The main difficulty is due to the right hand side f(x)/u gamma, since u = 0 on the boundary. In order to overcome this "obstacle", we approach our above model problem thanks to the smooth Dirichlet problems [GRAPHICS] and we prove that there exists a solution u as limit (in a suitable topology) of the sequence {u(n)}. To be more precise, we prove that the above model problem has a suitable solution u for every f in L-1(Omega) and for every gamma > 0 and how the regularity of u depends on the summability of f, on p and on gamma.