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

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