  We prove Liouville type theorems for $p$-harmonic functions on exterior
domains of the $d$-dimensional Euclidean space, where $1<p<\infty$ and $d\geq
2$. We show that every positive $p$-harmonic function satisfying zero
Dirichlet, Neumann or Robin boundary conditions and having zero limit as $|x|$
tends to infinity is identically zero. 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.

Analysis of PDEs