  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 $M$.
