# Weak Solution of a Singular Semilinear Elliptic Equation in a Bounded Domain

### Robert Dalmasso

Equipe EDP, Grenoble, France

## Abstract

We study the singular semilinear elliptic equation ∆_u_ + *f*(., *u*) = 0 in *D'*(Ω), where Ω ⊂ ℝ_n_ (*n* ≥ 1) is a bounded domain of class _C_1,1. *f* : Ω × (0, ∞) → [0, ∞) is such that *f*(., *u*) ∈ _L_1(Ω) for *u* > 0 and *u* → *f*(*x*, *u*) is continuous and nonincreasing for a.e. *x* in Ω. We assume that there exists a subset Ω ⊂ Ω with positive measure such that *f*(*x*, *u*) > 0 for *x* ∈ Ω and *u* > 0 and that ∫Ω *f*(*x*, *cd*(*x*, ∂Ω)) *dx* < ∞ for all c > 0. Then we show that there exists a unique solution u in *W_01,1(Ω) such that ∆_u* ∈ _L_1(Ω), *u* > 0 a.e. in Ω.