Cheng-Yau type gradient estimates for ${\mathrm{\Delta }}_f{v}^{\tau }+\lambda (x)v^l=0$ on smooth metric measure spaces

In this paper, by using the Saloff-Coste Sobolev-type inequality and Nash-Moser iteration, we proved a local gradient estimate of Cheng-Yau type for positive solutions to the equation

$${\mathrm{\Delta }}_f{v}^{\tau }+\lambda (x)v^l=0$$

on metric measure spaces with $m$-Bakry-Emery Ricci curvature bounded from below. Here $\tau >0$ and $l$ were constants, and $\lambda (x)$ was allowed to change sign. As applications, we also obtained a Liouville-type result and Harnack's inequality. Compared with previous works, this paper did not need to suppose the positive solutions are bounded and extended the ranges of $\tau$ and $l$.