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In this paper, we consider nonlinear equations involving the fractionalp-Laplacian $$ (-\lap)_p^s u(x)) \equiv C_{n,s,p} PV \int_{\mathbb{R}^n}\frac{|u(x)-u(y)|^{p-2}[u(x)-u(y)]}{|x-z|^{n+ps}} dz= f(x,u).$$  We prove a {\em maximum principle for anti-symmetric functions} and obtainother key ingredients for carrying on the method of moving planes, such as {\ema key boundary estimate lemma}. Then we establish radial symmetry andmonotonicity for positive solutions to semilinear equations involving thefractional p-Laplacian in a unit ball and in the whole space. We believe thatthe methods developed here can be applied to a variety of problems involvingnonlinear nonlocal operators.

Maximum principles for the fractional p-Laplacian and symmetry of solutions