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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 differentfrom the previous ones in deriving Liouville theorems.  For any given hyper-plane in $\mathbb{R}^n$, we show that $u$ is symmetricabout the plane. To this end, we established a {\em maximum principle} foranti-symmetric functions on any half space. We believe that this {\em maximumprinciple}, as well as the ideas in the proof, will become useful tools instudying a variety of problems involving nonlinear non-local operators.

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