Souplet-Zhang and Hamilton type gradient estimates for nonlinear elliptic equations on smooth metric measure spaces

In this article we present new gradient estimates for positive solutions to a class of nonlinear elliptic equations involving the $f$-Laplacian on a smooth metric measure space. The gradient estimates of interest are of Souplet Zhang and Hamilton types respectively and are established under natural lower bounds on the generalised Bakry-\'Emery Ricci curvature tensor. From these estimates we derive amongst other things Harnack inequalities and general global constancy and Liouville-type theorems. The results and approach undertaken here provide a unified treatment and extend and improve various existing results in the literature. Some implications and applications are presented and discussed.