On the uniqueness of solutions of an nonlocal elliptic system

2014/03/10 by Kelei Wang, Juncheng Wei, Wang, Kelei +1
Computer Science · Mathematics · #35B45 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1403.2346

openalex publication_date 2014/03/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the following elliptic system with fractional laplacian -(-Δ)^su=uv², -(-Δ)^sv=vu², u,vgt;0 on \Rⁿ, where s∈(0,1) and (-Δ)^s is the s-Lapalcian. We first prove that all positive solutions must have polynomial bound. Then we use the Almgren monotonicity formula to perform a blown-down analysis to s-harmonic functions. Finally we use the method of moving planes to prove the uniqueness of the one dimensional profile, up to translation and scaling.

Citations

• On Phase-Separation Models: Asymptotics and Qualitative Properties
• Strong competition versus fractional diffusion: the case of\n Lotka-Volterra interaction
• The Dirichlet problem for the fractional Laplacian: regularity up to the\n boundary

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