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

Abstract

In this paper, we consider the nonlinear elliptic equation $$\hat{I}{”}_f{v}^{\ddot{I}}„+\hat{I}»v=0$$ on a complete smooth metric measure space with $m$-Bakry-Émery Ricci curvature bounded from below, where $\ddot{I}„>0$ and $\hat{I}»$ 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).

22 pages