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

Original · EN

In this paper, we consider nonlinear equations involving the fractional p-Laplacian (-)ₚˢ u(x)) ≡ Cₙ,ₛ,ₚ PV ∫ᵣₙ |u(x)-u(y)|ᵖ⁻²[u(x)-u(y)]|x-z|n+ps dz= 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 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.