A note on Liouville type theorem of elliptic inequality $\Delta u+u^\sigma\leq 0$ on Riemannian manifolds

TL;DR

This paper improves existing results on the uniqueness of nonnegative solutions to a specific elliptic inequality on Riemannian manifolds by establishing an integral volume growth condition, inspired by criteria for manifold parabolicity.

Contribution

It introduces an integral volume growth condition that guarantees uniqueness of solutions, extending previous pointwise bounds to a broader integral framework.

Findings

Integral volume growth condition ensures solution uniqueness

Extension of Grigor$^{ ext{'}$yan and Sun's results

Inspired by Varopoulos-Grigor$^{ ext{'}$yan's parabolicity criterion

Abstract

Let $\sigma>1$ and let $M$ be a complete Riemannian manifold. In a recent work [9], Grigor$^{\prime}$yan and Sun proved that a pointwise upper bound of volume growth is sufficient for uniqueness of nonnegative solutions of elliptic inequality $$(*)\quad\qquad\qquad\qquad \Delta u(x)+u^\sigma(x)\leq 0,\qquad x\in M.\quad\qquad \qquad\qquad $$ In this note, we improve their result to that an \emph{integral condition} on volume growth implies the same uniqueness of ($*$). It is inspired by the well-known Varopoulos-Grigor$^{\prime}$yan's criterion for parabolicity of $M$.