Communications in Contemporary Mathematics ( IF 1.278 ) Pub Date : 2021-02-19 , DOI: 10.1142/s021919972150005x
Wei Dai; Zhao Liu; Pengyan Wang

In this paper, we are concerned with the following Dirichlet problem for nonlinear equations involving the fractional $p$-Laplacian: 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 $(−Δ)pαu(x)=Cn,α,pP.V.∫ℝn|u(x)−u(y)|p−2[u(x)−u(y)]|x−y|n+αpdy.$ 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].

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