Existence and Global Asymptotic Behavior of Singular Positive Solutions for Radial Laplacian

Abstract

The aim of this paper is to establish existence and uniqueness of a positive continuous solution to the following singular nonlinear problem. {-t1-ntn-1u′′=a(t)uσ, t∈(0,1), limt→0⁡tn-1u′(t)=0, u(1)=0}, where n≥3,σ<1, and a denotes a nonnegative continuous function that might have the property of being singular at t=0 and /or t=1 and which satisfies certain condition associated to Karamata class. We emphasize that the nonlinearity might also be singular at u=0, while the solution could blow-up at 0. Our method is based on the global estimates of potential functions and the Schauder fixed point theorem.