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

We prove Liouville type theorems for $p$-harmonic functions on exterior domains of the $d$-dimensional Euclidean space, where $1<p<\mathrm{\infty }$ and $d\geq 2$\mathrm{.}Weshowthateverypositive$p$-harmonic function satisfying zero Dirichlet, Neumann or Robin boundary conditions and having zero limit as $\mathrm{\mid }x\mathrm{\mid }$ 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.