On a Dirichlet problem in bounded domains with singular nonlinearity

In this paper we prove the existence and regularity of positive solutions of the homogeneous Dirichlet problem −∆u = g(x, u) in Ω, u = 0 on ∂ Ω, where g(x, u) can be singular as u → 0+ and 0 ≤ g(x, u) ≤ φ0 (x) up or 0 ≤ g(x, u) ≤ φ0 (x)(1 + 1/u^p ), with φ_0∈ L^m(Ω), 1≤ m. There are no assumptions on the monotonicity of g(x, ·) and the existence of super- or sub-solutions