Gradient estimates for $(p,V)$-harmonic functions on Riemannian manifolds

TL;DR

This paper derives explicit global gradient estimates for positive $(p,V)$-harmonic functions on complete Riemannian manifolds, utilizing volume comparison, Sobolev embedding, and Moser iteration under Bakry-Émery curvature conditions.

Contribution

It introduces new gradient estimates for $(p,V)$-harmonic functions on manifolds with Bakry-Émery curvature, expanding understanding of their behavior.

Findings

Established volume comparison and Sobolev embedding theorems under Bakry-Émery curvature.

Derived explicit global gradient estimates for positive $(p,V)$-harmonic functions.

Applied Moser iteration method to obtain key estimates.

Abstract

In this paper, we study $(p,V)$-harmonic functions on complete Riemannian manifolds using the Moser iteration method. A volume comparison theorem and a Sobolev embedding theorem are established under the Bakry-$\acute{E}$mery curvature condition. Moreover, we obtain an explicit global gradient estimate for positive entire $(p,V)$-harmonic functions.