Universal gradient estimates for solutions of $\Delta_{p,f}u+au^{\sigma}\ln u=0$ on complete Riemannian manifolds

TL;DR

This paper derives sharp gradient estimates and Liouville theorems for solutions of a weighted p-Laplacian equation with a logarithmic nonlinearity on complete Riemannian manifolds with curvature bounds.

Contribution

It introduces new gradient estimates for a nonlinear PDE on Riemannian manifolds, extending Liouville theorems under curvature conditions.

Findings

Established sharp gradient bounds for solutions.

Proved Liouville theorems for the equation.

Applied Nash-Moser iteration technique.

Abstract

In this paper, we consider the weighted $p$-Laplacian equation $$ \Delta_{p,f}u+au^{\sigma}\ln u=0$$ defined on a complete smooth metric measure space under the conditon that the $m$-Bakry-\'{E}mery Ricci curvature has a lower bound, where $a$, $\sigma$ are two nonzero real constants. By applying the Nash-Moser iteration, we obtain sharp gradient estimates and thereby establish Liouville theorems for the above equation.