Sublinear elliptic problems with a Hardy potential

TL;DR

This paper investigates positive solutions to sublinear elliptic equations with Hardy potentials, providing a comprehensive analysis of radial solutions and establishing existence results using ODE techniques and Hardy constants.

Contribution

It offers a detailed classification of radial solutions and introduces methods to construct solutions in general domains using Hardy potentials and contraction operators.

Findings

Complete characterization of radial solutions

Existence of positive solutions in general domains

Use of Hardy constant as a key analytical tool

Abstract

In this paper we study the positive solutions of sub linear elliptic equations with a Hardy potential which is singular at the boundary. By means of ODE techniques a fairly complete picture of the class of radial solutions is given. Local solutions with a prescribed growth at the boundary are constructed by means of contraction operators. Some of those radial solutions are then used to construct ordered upper and lower solutions in general domains. By standard iteration arguments the existence of positive solutions is proved. An important tool is the Hardy constant.