An elliptic equation with singular nonlinearity

Abstract

The purpose of this paper is to study the problem (P) −Δu+u−α=f in Ω, u=0 on ∂Ω, u−α∈L1(Ω), u>0 in Ω, where Ω is a bounded smooth open set of RN, f≥0, f∈L1(Ω) and 0<α<1. When α≤0 this problem corresponds to a monotone semilinear equation for which the existence and uniqueness of solutions are well known. On the other hand, semilinear equations with nonmonotone perturbations have also been considered in the literature by many different methods. The originality of our study comes from the singularity of the nonlinearity in (P).