Cheng-Yau type gradient estimates for $ \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$ \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 $.