## 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.

: Primary 35B53,35J92,35B40.

This paper is available as a pdf (244kB) file.

 Friday, February 14, 2014