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

Abstract

We prove Liouville type theorems for $p$-harmonic functions on an exterior domain ${\mathbb{R}}^d$, where $1<p<\mathrm{\infty }$ and $d\ge 2$. If $1<p<d$ we show that every positive $p$-harmonic function satisfying zero Dirichlet, Neumann or Robin boundary conditions is constant. For $p\ge d$ and $p\ne 1$ we show that positive $p$-harmonic functions are either constant or behave asymptotically like the fundamental solution of the $p$-Laplace operator. In the case of zero Neumann boundary conditions, we establish that every semi-bounded $p$-harmonic function is constant.