A note on Liouville type theorem of elliptic inequality $\Delta u+u^{\sigma }\le 0$ 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 $$(\ast )\Delta u(x)+u^{\sigma }(x)\le 0,x\in M.$$ In this note, we improve their result to that an \emph{integral condition} on volume growth implies the same uniqueness of ($\ast$). It is inspired by the well-known Varopoulos-Grigor${}^{\prime }$yan's criterion for parabolicity of $M$.

Fixed typos and updated references, 8 pages