In this paper, we use the Nash-Moser iteration method to prove the local and global gradient estimates of positive solutions to the weighted p-Laplacian equation aligned , ₅ u+au^ =0 aligneddefined on a complete smooth metric measure space under the condition that the m-Bakry-Émery Ricci curvature has a lower bound, where p>1, a and R are constants and , ₅ u=e^fdiv (e^-f| u|^p-2 u) is the weighted p-Laplacian operator. As applications, we derive Liouville type theorems and Harnack inequalities for positive solutions to the above equation.
Huang et al. (Thu,) studied this question.