Nonlinear elliptic equations with singular nonlinearities

Abstract

In this paper we study nonlinear elliptic boundary value problems with singular nonlinearities whose simplest example is

−div (|∇u|^p−2∇u)=f/u^γ 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∈L^m(Ω), m≥1.

The main difficulty is due to the right hand side f(x)/u^γ, since u=0 on the boundary. In order to overcome this “obstacle”, we approach our above model problem thanks to the smooth Dirichlet problems

u_n∈W^1,p₀(Ω):

−div (|∇u_n|^p−2∇u_n)=min(f(x),n)/(|u_n|+1/n)^γ

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 L¹(Ω) and for every γ>0 and how the regularity of u depends on the summability of f, on p and on γ.