Gradient Estimates and Liouville Type Theorems for a Weighted Nonlinear p-Laplacian Equation on Compact Smooth Metric Measure Spaces

In this paper, we prove gradient estimates for positive smooth solutions to the weighted nonlinear p-Laplacian equation Delta(p,phi)u + au(p-1 )log u + lambda u(p-1) = 0 on compact smooth metric measure spaces with m-Bakry-<acute accent>Emery Ricci curvature bounded from below, where a, lambda and p > 1 are some given constants. We generalize and improve the previous results due to Ma and Zhu [Arch. Math. (Basel), 117 (2021)] on the manifold case since one extends the range of p and does not need to suppose non-negativity on the m-Bakry-<acute accent>Emery Ricci curvature. As applications, we also obtain some Liouville type results for the above equation.