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

2017/05/13 by Chen, Wenxiong, Li, Congming
#35J60 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1705.04891

Abstract

In this paper, we consider nonlinear equations involving the fractional p-Laplacian (-\lap)_p^s u(x)) ≡ C_n,s,p PV ∫_ℝⁿ \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 obtain other key ingredients for carrying on the method of moving planes, such as \em a key boundary estimate lemma. 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.

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