Radial symmetry of positive solutions involving the fractional Laplacian

2013/11/27 by Patricio Felmer, Ying Wang, Felmer, Patricio +1 · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #math.AP

paper · pdf · doi:10.48550/arxiv.1311.6952

27 pages, Communications in Contemporary Mathematics 2013

arxiv created 2013/11/27 · openalex publication_date 2013/11/27 · arxiv updated 2013/11/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The aim of this paper is to study radial symmetry and monotonicity properties for positive solution of elliptic equations involving the fractional Laplacian. We first consider the semi-linear Dirichlet problem (-Δ)^α u=f(u)+g, \rmin B₁, u=0 \rm in B₁^c, where (-Δ)^α denotes the fractional Laplacian, α∈(0,1), and B₁ denotes the open unit ball centered at the origin in \R^N with N≥2. The function f:[0,∞)→\R is assumed to be locally Lipschitz continuous and g: B₁→\R is radially symmetric and decreasing in |x|. In the second place we consider radial symmetry of positive solutions for the equation (-Δ)^α u=f(u), \rmin \R^N, with u decaying at infinity and f satisfying some extra hypothesis, but possibly being non-increasing. Our third goal is to consider radial symmetry of positive solutions for system of the form (-Δ)^α₁ u=f₁(v)+g₁, & \rmin B₁, [2mm] (-Δ)^α₂ v=f₂(u)+g₂, & \rmin B₁, [2mm] u=v =0, & \rmin B₁^c, where α₁,α₂∈(0,1), the functions f₁ and f₂ are locally Lipschitz continuous and increasing in [0,∞), and the functions g₁ and g₂ are radially symmetric and decreasing. We prove our results through the method of moving planes, using the recently proved ABP estimates for the fractional Laplacian. We use a truncation technique to overcome the difficulty introduced by the non-local character of the differential operator in the application of the moving planes.

Citations

• A symmetry problem in potential theory
• Regularity of the obstacle problem for a fractional power of the laplace operator
• Aleksandrov-Bakelman-Pucci Type Estimates For Integro-Differential Equations
• The Dirichlet problem for the fractional Laplacian: regularity up to the boundary

Cited by

• A note on spectral Monte-Carlo method for fractional Poisson equation on high-dimensional ball

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