A Singular ODE Related to Quasilinear Elliptic Equations

Abstract

We consider a quasilinear elliptic problem with the natural growth in the gradient. Existence, non-existence, uniqueness, and qualitative properties of positive solutions are obtained. We consider both weak and strong solutions. All results are based on the study of a suitable singular ODE of the first order. We also introduce a comparison principle for a class of nonlinear integral operators of Volterra type that enables to obtain uniqueness of weak solutions of the quasilinear equation.