Liouville type theorems for the $p$-harmonic functions on certain manifolds

Jingyi Chen and Yue Wang

Vol. 282 (2016), No. 2, 313–327

DOI: 10.2140/pjm.2016.282.313

Abstract

We show that for a certain range of $p>n$, the Dirichlet problem at infinity is unsolvable for the $p$-Laplace equation for any nonconstant continuous boundary data 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 ${ℝ}^2$ with positive Gauss curvature such that every positive $p$-harmonic function must be constant for $p\ge 4$.

Keywords

Liouville type theorems, $p$-harmonic functions, gradient Ricci solitons

Mathematical Subject Classification 2010

Primary: 53C21, 58J05

Milestones

Received: 13 November 2014

Revised: 3 September 2015

Accepted: 20 October 2015

Published: 3 March 2016