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

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:270

Article

Liu, Zhao^1,2 

[1] Jiangxi Sci & Technol Normal Univ, Sch Math & Comp Sci, Nanchang 330038, Jiangxi, Peoples R China

[2] Yeshiva Univ, Dept Math, New York, NY 10033 USA

关键词: Fractional p-Laplacians;    Maximum principles;    Unbounded domains;    Sliding methods;    Asymptotic behavior;    Monotonicity of solutions;    Uniqueness;   

DOI  :  10.1016/j.jde.2020.09.001

来源: Elsevier

【 摘 要 】

In this paper, we consider the following non-linear equations in unbounded domains Omega with exterior Dirichlet condition: {(-Delta u)(p)(s)u(x) = f (u(x)), x is an element of Omega, u(x) > 0, x is an element of Omega, u(x) = 0, x is an element of R-n\Omega, where (-Delta)(p)(s) is the fractional p-Laplacian defined as (-Delta)(p)(s)u(x) = C-n,C-s,C-p P.V. integral(Rn )vertical bar u(x) - u(y)vertical bar(p-2)[u(x) - u(y)]/vertical bar x - y vertical bar(n+sp) dy (0.1) 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 Omega. 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. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

Free