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 \(1< p<\infty \) and \(d\geq 2\). If \(1< p< d \) we show that every positive \(p\)-harmonic function satisfying zero Dirichlet, Neumann or Robin boundary conditions is constant. For \(p\geq d\) and \(p\neq 1\) 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.