Sublinear elliptic problems under radiality. Harmonic $NA$ groups and Euclidean spaces

TL;DR

This paper investigates sublinear elliptic equations on Euclidean spaces and harmonic $NA$ groups, establishing conditions for solutions' existence and proving a Harnack inequality under radiality assumptions.

Contribution

It provides necessary and sufficient conditions for entire bounded or large solutions of sublinear elliptic equations on Euclidean and harmonic $NA$ groups, with a focus on radiality.

Findings

Conditions for existence of solutions are characterized.

A Harnack-type inequality for positive solutions is established.

Results apply to Euclidean spaces and harmonic $NA$ groups.

Abstract

Let $\L $ be the Laplace operator on $\R ^d$, $d\geq 3$ or the Laplace Beltrami operator on the harmonic $NA$ group (in particular on a rank one noncompact symmetric space). For the equation $ \L u - \varphi(\cdot,u)=0$ we give necessary and sufficient conditions for the existence of entire bounded or large solutions under the hypothesis of radiality of $\varphi$ with respect to the first variable. A Harnack-type inequality for positive continuous solutions is also proved.