Positive Solutions for a Singular Nonlinear Problem on a Bounded Domain in R 2

For a bounded regular Jordan domain Ω in R 2, we introduce and study a new class of functions K(Ω) related on its Green function G. We exploit the properties of this class to prove the existence and the uniqueness of a positive solution for the singular nonlinear elliptic equation Δu+ϕ(x,u)=0, in D′(Ω), with u=0 on ∂Ω and u∈C―(Ω), where ϕ is a nonnegative Borel measurable function in Ω×(0,∞) that belongs to a convex cone which contains, in particular, all functions ϕ(x,t)=q(x)t −γ,γ>0 with nonnegative functions q∈K(Ω). Some estimates on the solution are also given.