A Note on Liouville Type Theorem of Elliptic Inequality Δu + u σ ⩽ 0 on Riemannian Manifolds

Abstract

Let σ > 1 and let M be a complete Riemannian manifold. In a very recent work (Grigor′yan and Sun 2014), Grigor ′yan and Sun proved that a Liouville type theorem holds for nonnegative solutions of elliptic inequality $$ {\Delta} u(x)+u^{\sigma}(x)\leqslant 0,\qquad x\in M. $$ (*) via a pointwise condition of volume growth of geodesic balls. In this note, we improve their result showing that an integral condition on volume growth implies the same uniqueness of solutions to Eq. (*). It is inspired by the well-known Varopoulos-Grigor′yan’s criterion for parabolicity of M.