Local and global gradient estimates for Finsler $p$-harmonic functions

In this paper, we give the local and global gradient estimates for positive Finsler $p$‑eigenfunctions on a complete Finsler manifold $M$ with the weighted Ricci curvature bounded from below by a negative constant. As applications, we obtain some Liouville and Harnack theorems, and the global gradient estimates for positive Finsler $p$‑harmonic functions. As a by-product of the global estimate, we obtain an upper bound of the first $p$‑eigenvalue $\lambda_{1,p}$ for Finsler $p$-Laplacian $\Delta_p$. Further, we study the geometric structure at infinity of Finsler manifolds with $\lambda_{1,p}$ achieving its maximum value.

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The author was supported by NNSFC (Nos. 12071423, 11671352) and Scientific Research Fundation of HDU (No. KYS075621060).

Received 24 June 2018

Accepted 9 August 2019

Published 29 November 2022