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

preprint (PDF), 28 January 2014
Positivity 19 (2015), 577–586
Original version appears at DOI 10.1007/s11117-014-0316-2
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.