A Liouville theorem for \(p\)-harmonic functions on exterior domains

E. N. Dancer, Daniel Daners, Daniel Hauer

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 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.

Keywords: elliptic boundary-value problems, Liouville-type theorems, \(p\)-Laplace operator, \(p\)-harmonic functions, exterior domain.

AMS Subject Classification: Primary 35B53,35J92,35B40.