Local and Global Gradient Estimates for p-Laplace Equations on Complete Smooth Metric Measure Spaces and Liouville Type Theorems

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 $$\begin{aligned} \Delta _{p,f} u+au^\sigma =0 \end{aligned}$$defined 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 $$\sigma \in \mathbb {R}$$ are constants and $$\Delta _{p,f} u=e^{f}\textrm{div}(e^{-f}|\nabla u|^{p-2}\nabla u)$$ is the weighted p-Laplacian operator. As applications, we derive Liouville type theorems and Harnack inequalities for positive solutions to the above equation.