Cheng-Yau logarithmic gradient estimates for a nonlinear elliptic equation on smooth metric measure spaces

TL;DR

This paper derives new gradient estimates for positive solutions of a nonlinear elliptic equation on smooth metric measure spaces, leading to Liouville theorems and Harnack inequalities, thus extending previous results in the field.

Contribution

It introduces improved local and global gradient estimates for nonlinear elliptic equations on smooth metric measure spaces, generalizing prior work by Wang and Zhao.

Findings

New local gradient estimates for solutions

Liouville type theorem established

Harnack inequality proved

Abstract

In this paper, we consider the nonlinear elliptic equation $$\Delta_fv^\tau+\lambda v=0$$ on a complete smooth metric measure space with $m$-Bakry-\'{E}mery Ricci curvature bounded from below, where $\tau>0$ and $\lambda$ are constant. We obtain some new local gradient estimates for positive solutions to the equation using the Nash-Moser iteration technique. As applications of these estimates, we obtain a Liouville type theorem and a Harnack inequality, and the global gradient estimates for such solutions. Our results generalize and improve the estimates in Wang (J. Differential Equations 260:567-585, 2016) and Zhao (Arch. Math. (Basel) 114:457-469, 2020).