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

We prove a maximum principle for anti-symmetric functions and obtain other key ingredients for carrying on the method of moving planes, such as a variant of the Hopf Lemma - a boundary estimate lemma which plays the role of the narrow region principle. Then we establish radial symmetry and monotonicity for positive solutions to semilinear equations involving the fractional p-Laplacian in a unit ball and in the whole space. We believe that the methods developed here can be applied to a variety of problems involving nonlinear nonlocal operators.