Liouville type theorems for the p-harmonic functions

TL;DR

This paper establishes Liouville type theorems for p-harmonic functions on specific manifolds, showing the nonexistence of nonconstant solutions under certain conditions related to the geometry and the parameter p.

Contribution

It demonstrates the unsolvability of the Dirichlet problem at infinity for p-Laplace equations on particular Cartan-Hadamard manifolds and constructs examples of incomplete metrics with constant p-harmonic functions.

Findings

Dirichlet problem at infinity is unsolvable for certain p>n on specific manifolds.

Constructs an incomplete Riemannian metric with positive curvature where positive p-harmonic functions are constant for p≥4.

Shows that geometric properties influence the behavior of p-harmonic functions.

Abstract

We show that the Dirichlet problem at infinity is unsolvable for the p-Laplace equation for any nonconstant continuous boundary data, for certain range of p>n, on an n-dimensional Cartan-Hadamard manifold constructed from a complete noncompact shrinking gradient Ricci soliton. Using the steady gradient Ricci soliton, we find an incomplete Riemannian metric on ${\mathbb R}^2$ with positive Gauss curvature such that every positive p-harmonic function must be constant for $p\geq 4$.