A Liouville theorem for $p$-harmonic functions on exterior domains

TL;DR

This paper establishes Liouville theorems for p-harmonic functions on exterior domains, showing under various boundary conditions and growth constraints that such functions must be trivial or have specific asymptotic behavior.

Contribution

It proves new Liouville type results for p-harmonic functions in exterior domains, including conditions for triviality and asymptotic behavior based on boundary conditions and parameter ranges.

Findings

Positive p-harmonic functions with zero boundary conditions and zero limit at infinity are identically zero.

Semi-bounded p-harmonic functions are constant for 1<p<d under Neumann boundary conditions.

For p≥d, p-harmonic functions are either constant or asymptotic to the fundamental solution.

Abstract

We prove Liouville type theorems for $p$-harmonic functions on exterior domains of the $d$-dimensional Euclidean space, where $1<p<\infty$ and $d\geq 2$. We show that every positive $p$-harmonic function satisfying zero Dirichlet, Neumann or Robin boundary conditions and having zero limit as $|x|$ tends to infinity is identically zero. In the case of zero Neumann boundary conditions, we establish that any semi-bounded $p$-harmonic function is constant if $1<p<d$. If $p\ge d$, then it is either constant or it behaves asymptotically like the fundamental solution of the homogeneous $p$-Laplace equation.