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

In this paper, we consider nonlinear equations involving the fractional p-Laplacian (-Delta)(p)(s)u(x)) (math) C-n,C-sp PV integral(Rn)vertical bar u(x) - u(Y)vertical bar(P-2)[u(x) - u(y)]/vertical bar x - y vertical bar(n+sp)dy = f(x, u). 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. (C) 2018 Elsevier Inc. All rights reserved.