Positive solutions for elliptic equations with singular nonlinearity

Abstract

We study an elliptic boundary-value problem with singular non-linearity via the method of monotone iteration scheme: -∆u(x) = ƒ(x, u(x)), x ∈ Ω, u(x) = ϕ(x), x ∈ ∂Ω, Where ∆ is the Laplacian operator, Ω is a bounded domain in ℝN, N ≥ 2, ϕ ≥ 0 may take the value 0 on ∂Ω, and ƒ(x, s) is possibly singular near s = 0. We prove the existence and the uniqueness of positive solutions under a set of hypotheses that do not make neither monotonicity nor strict positivity assumption on ƒ(x, s), which improvements of some previous results.