Liouville theorems for nonlinear elliptic equations on Riemannian manifolds

Abstract

Abstract Motivated by the joint work of the first author with Khanh and Ngo, and a recent work regarding to a Liouville property of p-Lichnerowicz equation by Zhao, in this paper, we study the following elliptic equation on smooth metric measure spaces ( M , g , e − f d v ) with a weighted Poincare inequality Δ p , f v + h ( v ) = 0 . We obtain a Liouville type theorem for this equation by using the approach in [11] by the first author and Seo. As a consequence, we are able to point out a Liouville result which is a refinement and improvement of those by Zhao [31] .