Gradient estimates for weighted $p$-Laplacian equations on Riemannian manifolds with a Sobolev inequality and integral Ricci bounds

TL;DR

This paper establishes local gradient estimates for weighted p-Laplacian equations on Riemannian manifolds with Sobolev inequalities and small integral Ricci curvature, leading to Liouville type results and new gradient bounds.

Contribution

It provides novel local gradient estimates under integral Ricci bounds and Sobolev inequalities, extending previous results to more general geometric settings.

Findings

Proved local gradient estimates for weighted p-Laplacian equations.

Derived Liouville type theorems under Ricci curvature bounds.

Established new gradient bounds assuming small integral Ricci curvature.

Abstract

In this paper, we consider the non-linear general $p$-Laplacian equation $\Delta_{p,f}u+F(u)=0$ for a smooth function $F$ on smooth metric measure spaces. Assume that a Sobolev inequality holds true on $M$ and an integral Ricci curvature is small, we first prove a local gradient estimate for the equation. Then, as its applications, we prove several Liouville type results on manifolds with lower bounds of Ricci curvature. We also derive new local gradient estimates provided that the integral Ricci curvature is small enough.