Nonlinear elliptic equations with singular nonlinearities

In this paper we study nonlinear elliptic boundary value problems with singular nonlinearities whose simplest example is{div( |δu|p2δu ) = f u in ω, u =0 on ω, where ω is a bounded open set in R N (N ≥ 2),γ >0, 1 < p < N, 0 ≤ f ε L m(ω), m ≥ 1. The main difficulty is due to the right hand side f(x)/uγ , since u = 0 on the boundary. In order to overcome this "obstacle", we approach our above model problem thanks to the smooth Dirichlet problems { un ε W 1,p 0 (ω): div|δun|p2δun ) = min(f(x), n) (|un| + 1/n)γ and we prove that there exists a solution u as limit (in a suitable topology) of the sequence {un}. To be more precise, we prove that the above model problem has a suitable solution u for every f in L1(ω) and for every γ > 0 and how the regularity of u depends on the summability of f, on p and on γ. Keywords: nonlinear elliptic equations, singular elliptic equations, quasilinear elliptic equations with p-Laplacian.