@article{Zeng2025, 
author = {Fanqi Zeng and Cheng Jin and Peilong Dong and Xinying Jiang},
title = {Cheng-Yau type gradient estimates for        Δ    f        v          τ        +  λ  (  x  )      v    l    =  0 on smooth metric measure spaces},
year = {2025},
journal = {Electronic Research Archive},
volume = {33},
number = {7},
pages = {4307-4326},
keywords = {nonlinear elliptic equation, gradient estimate, Liouville theorem, Harnack inequality},
url = {https://www.sciopen.com/article/10.3934/era.2025195},
doi = {10.3934/era.2025195},
abstract = {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         Δ    f        v          τ        +  λ  (  x  )      v    l    =  0on metric measure spaces with    m-Bakry-Emery Ricci curvature bounded from below. Here    τ  >  0 and    l were constants, and    λ  (  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    τ and    l.}
}