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) p α u = f (x , u , ∇ u) , u > 0 in  Ω , u ≡ 0 in  ℝ n ∖ Ω , where Ω is a bounded or an unbounded domain which is convex in x 1 -direction, and (− Δ) p α is the fractional p -Laplacian operator defined by (− Δ) p α u (x) = C n , α , p P. V. ∫ ℝ n | u (x) − u (y) | p − 2 [ u (x) − u (y) ] | x − y | n + α p d y. 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]. [ABSTRACT FROM AUTHOR]