Gradient estimates for a nonlinear elliptic equation on smooth metric measure spaces and applications

Abstract

In this paper local and global gradient estimates are obtained for positive solutions to the following nonlinear elliptic equationΔfu+p(x)u+q(x)uα=0, on complete smooth metric measure spaces (MN,g,e−fdv) with ∞-Bakry-Émery Ricci tensor bounded from below, where α is an arbitrary real constant, p(x) and q(x) are smooth functions. As an application, Liouville-type theorems for various special cases of the equation are recovered. Furthermore, we discuss nonexistence of smooth solution to Yamabe type problem on (MN,g,e−fdv) with nonpositive weighted scalar curvature.