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

In this paper local and global gradient estimates are obtained for positive solutions to the following nonlinear elliptic equation [Formula: see text] on complete smooth metric measure spaces [Formula: see text] with ∞-Bakry-Émery Ricci tensor bounded from below, where α is an arbitrary real constant, [Formula: see text] and [Formula: see text] 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 [Formula: see text] with nonpositive weighted scalar curvature.