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

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.