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.