Monotonicity and symmetry of positive solutions to fractional p -Laplacian equation

In this paper, we are concerned with the following Dirichlet problem for nonlinear equations involving the fractional p-Laplacian: (-Δ)pαu = f(x,u,u),u > 0in ω,u 0 in Rnω, where ω is a bounded or an unbounded domain which is convex in x1-direction, and (-Δ)pα is the fractional p-Laplacian operator defined by (-Δ)pαu(x) = C n,α,pP.V.∫Rn|u(x) - u(y)|p-2[u(x) - u(y)] |x - y|n+αp dy. Under some mild assumptions on the nonlinearity f(x,u,u), we establish the monotonicity and symmetry of positive solutions to the nonlinear equations involving the fractional p-Laplacian in both bounded and unbounded domains. Our results are extensions of Chen and Li [Maximum principles for the fractional p-Laplacian and symmetry of solutions, Adv. Math. 335 (2018) 735-758] and Cheng et al. [The maximum principles for fractional Laplacian equations and their applications, Commun. Contemp. Math. 19(6) (2017) 1750018].