Maximum principles for a fully nonlinear fractional order equation and symmetry of solutions

2016/04/16 by Wenxiong Chen, Congming Li, Chen, Wenxiong +3
Mathematics · #35J60 #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1604.04806

openalex publication_date 2016/04/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we consider equations involving fully nonlinear nonlocal operators F_α(u(x)) ≡ C_n,α PV ∫_ℝⁿ \fracG(u(x)-u(z))|x-z|^n+α dz= f(x,u). We prove a maximum principle and obtain key ingredients for carrying on the method of moving planes, such as narrow region principle and decay at infinity. Then we establish radial symmetry and monotonicity for positive solutions to Dirichlet problems associated to such fully nonlinear fractional order equations in a unit ball and in the whole space, as well as non-existence of solutions on a half space. We believe that the methods develop here can be applied to a variety of problems involving fully nonlinear nonlocal operators. We also investigate the limit of this operator as α→ 2 and show that F_α(u(x)) → a(-Δu(x)) + b |\bigtriangledown u(x)|² .

Citations

• Regularity of the obstacle problem for a fractional power of the laplace operator
• A Liouville Theorem for the Fractional Laplacian
• Uniqueness of Radial Solutions for the Fractional Laplacian

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