Local and global estimates for solutions to $\Delta_{p}v+av^{\sigma}+rv^{\tau}=0$ on a complete Finsler space and their applications, II

Hongmei Zhu and Wenhao Wang

Vol. 345 (2026), No. 2, 431–475

DOI: 10.2140/pjm.2026.345.431

Abstract

We investigate the local and global properties of positive solutions to the Finsler $p$-Laplacian equation

$${\mathrm{\Delta }}_{p}v+a{v}^{\sigma }+r{v}^{\tau }=0$$

on a complete noncompact Finsler measure space with the weighted Ricci curvature ${\mathrm{Ric}<!--FUNCTION\; APPLICATION-->}_N$ bounded below, where $a,r\in ℝ$, $p>1$ and $0\le \sigma \le \tau <p^{\ast }-1$ are real constants, where $p^{\ast }=\frac{np}{n-p}$ is the Sobolev duality of $p$. Under certain conditions on $p$, $a$, $r$, $\sigma$ and $\tau$, we obtain Cheng–Yau-type gradient estimates for positive solutions. These gradient estimates allow us to derive some Liouville-type theorems and Harnack inequalities. Based on the local gradient estimates obtained, we also obtain the explicit global gradient estimates for such entire positive solutions.

Keywords

Finsler metric, gradient estimates, Harnack inequality

Mathematical Subject Classification

Primary: 53B40

Secondary: 53C60

Milestones

Received: 17 May 2026

Revised: 12 August 2026

Accepted: 17 August 2026

Published: 26 October 2026