Nonexistence of positive supersolutions of elliptic equations via the maximum principle

TL;DR

This paper presents a new, simple maximum principle-based method to prove the nonexistence of positive supersolutions for a wide class of elliptic inequalities in unbounded domains, including nonlinear and fully nonlinear operators.

Contribution

It introduces a robust and versatile maximum principle approach applicable to various elliptic inequalities and systems, providing new optimal nonexistence results.

Findings

Applicable to quasilinear and fully nonlinear operators

Works in entire space, exterior, and cone-like domains

Yields new optimal nonexistence conditions

Abstract

We introduce a new method for proving the nonexistence of positive supersolutions of elliptic inequalities in unbounded domains of $\mathbb{R}^n$. The simplicity and robustness of our maximum principle-based argument provides for its applicability to many elliptic inequalities and systems, including quasilinear operators such as the $p$-Laplacian, and nondivergence form fully nonlinear operators such as Bellman-Isaacs operators. Our method gives new and optimal results in terms of the nonlinear functions appearing in the inequalities, and applies to inequalities holding in the whole space as well as exterior domains and cone-like domains.