A maximum principle on unbounded domains and a Liouville theorem for fractional p-harmonic functions

In this paper, we establish the following Liouville theorem for fractional \emph{p}-harmonic functions. {\em Assume that $u$ is a bounded solution of $$(-\lap)^s_p u(x) = 0, \;\; x \in \mathbb{R}^n,$$ with $0<s<1$ and $p \geq 2$. Then $u$ must be constant.} A new idea is employed to prove this result, which is completely different from the previous ones in deriving Liouville theorems. For any given hyper-plane in $\mathbb{R}^n$, we show that $u$ is symmetric about the plane. To this end, we established a {\em maximum principle} for anti-symmetric functions on any half space. We believe that this {\em maximum principle}, as well as the ideas in the proof, will become useful tools in studying a variety of problems involving nonlinear non-local operators.