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

In this article, we present new gradient estimates for positive solutions to a class of non-linear 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–Émery 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.