Large versus bounded solutions to sublinear elliptic problems

TL;DR

This paper investigates the existence of bounded and large solutions to sublinear elliptic equations involving a second order elliptic operator, establishing conditions under which bounded solutions imply the non-existence of large solutions.

Contribution

It provides new conditions linking the existence of bounded solutions to the non-existence of large solutions for sublinear elliptic problems with Kato class nonlinearities.

Findings

Existence of bounded solutions implies no large solutions under general conditions.

The results apply to elliptic operators on possibly unbounded domains.

The analysis covers nonlinearities in the Kato class with sublinear growth.

Abstract

Let $L $ be a second order elliptic operator with smooth coefficients defined on a domain $\Omega \subset \mathbb{R}^d$ (possibly unbounded), $d\geq 3$. We study nonnegative continuous solutions $u$ to the equation $L u(x) - \varphi (x, u(x))=0$ on $\Omega $, where $\varphi $ is in the Kato class with respect to the first variable and it grows sublinearly with respect to the second variable. Under fairly general assumptions we prove that if there is a bounded non zero solution then there is no large solution.