Gradient and Hessian estimates for an elliptic equation on smooth metric measure spaces

Abstract

In this paper, we prove gradient estimates and Hessian estimates for positive solutions to the second-order elliptic equation Δfu=pua on a smooth metric measure space. As applications, we obtain some f-harmonic function behaviors and Liouville theorems on the (m,n)-quasi-Einstein metric. Moreover, we derive Shi's derivative estimates.