Maximum principles and monotonicity of solutions for fractional p-equations in unbounded domains

Abstract

In this paper, we consider the following non-linear equations in unbounded domains Ω with exterior Dirichlet condition:{(−Δ)psu(x)=f(u(x)),x∈Ω,u(x)>0,x∈Ω,u(x)=0,x∈Rn∖Ω, where (−Δ)ps is the fractional p-Laplacian defined as(0.1)(−Δ)psu(x)=Cn,s,pP.V.∫Rn|u(x)−u(y)|p−2[u(x)−u(y)]|x−y|n+spdy with 0<s<1 and p≥2.We first establish a maximum principle in unbounded domains involving the fractional p-Laplacian by estimating the singular integral in (0.1) along a sequence of approximate maximum points. Then, we obtain the asymptotic behavior of solutions far away from the boundary. Finally, we develop a sliding method for the fractional p-Laplacians and apply it to derive the monotonicity and uniqueness of solutions.There have been similar results for the classical Laplacian [3] and for the fractional Laplacian [39], which are linear operators. Unfortunately, many approaches there no longer work for the fully non-linear fractional p-Laplacian here. To circumvent these difficulties, we introduce several new ideas, which enable us not only to deal with non-linear non-local equations, but also to remarkably weaken the conditions on f(⋅) and on the domain Ω.We believe that the new methods developed in our paper can be widely applied to many problems in unbounded domains involving non-linear non-local operators.