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

TL;DR

This paper develops new gradient estimates of Souplet-Zhang and Hamilton types for positive solutions to nonlinear elliptic equations involving the f-Laplacian on smooth metric measure spaces, leading to Harnack inequalities and Liouville theorems.

Contribution

It introduces unified gradient estimates under Bakry-Émery curvature bounds, extending and improving previous results for nonlinear elliptic equations on metric measure spaces.

Findings

Established new gradient estimates of Souplet-Zhang and Hamilton types.

Derived Harnack inequalities and Liouville-type theorems from these estimates.

Provided applications and a unified approach to existing results.

Abstract

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.