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

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 $M$.