Strong solutions for singular Dirichlet elliptic problems

Abstract

We prove an existence result for strong solutions u ∈ W 2 , q ( Ω ) of singular semilinear elliptic problems of the form − Δ u = g ( ⋅ , u ) in Ω , u = τ on ∂ Ω , where 1 < q < ∞ , Ω is a bounded domain in R n with C 2 boundary, 0 ≤ τ ∈ W 2 − 1 q , q ( ∂ Ω ) , and with g : Ω × ( 0 , ∞ ) → [ 0 , ∞ ) belonging to a class of nonnegative Carathéodory functions, which may be singular at s = 0 and also at x ∈ S for some suitable subsets S ⊂ Ω ¯ . In addition, we give results concerning the uniqueness and regularity of the solutions. A related problem on punctured domains is also considered.