Positive solutions for a singular nonlinear problem on a bounded domain in R-2

摘要

For a bounded regular Jordan domain Omega in R-2, we introduce and study a new class of functions K(Omega) 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 Deltau + phi(x, u)=0, in D' (Omega), with u=0 on and uis an element ofC((&UOmega;) over bar),where phi is a nonnegative Borel measurable function in Omega x (0, infinity) that belongs to a convex cone which contains, in particular, all functions phi(x, t)=q(x)t(-gamma), gamma>0 with nonnegative functions qis an element ofK(Omega). Some estimates on the solution are also given. [References: 12]