Nonexistence results for elliptic differential inequalities with a potential on Riemannian manifolds

TL;DR

This paper studies conditions under which nonnegative solutions do not exist for elliptic differential inequalities with potentials on Euclidean spaces and Riemannian manifolds, emphasizing the influence of geometry and potential behavior at infinity.

Contribution

It provides new nonexistence results for elliptic inequalities on Riemannian manifolds considering geometric and potential asymptotic effects.

Findings

Nonexistence results depend on manifold geometry.

Behavior of potential at infinity influences solution existence.

Results extend classical Euclidean inequalities to Riemannian settings.

Abstract

In this paper we are concerned with a class of elliptic differential inequalities with a potential both on $\erre^m$ and on Riemannian manifolds. In particular, we investigate the effect of the geometry of the underlying manifold and of the behavior of the potential at infinity on nonexistence of nonnegative solutions.