Bubbling solutions for nonlocal elliptic problems

2014/10/20 by Juan Dávila, Dávila, Juan, Luis Fernando López Ríos +3
Computer Science · Mathematics · #35B20 #35B33 #35B38 #35J91 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1410.5461

openalex publication_date 2014/10/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We investigate bubbling solutions for the nonlocal equation A_Ω^s u =u^p, u gt;0 in Ω, under homogeneous Dirichlet conditions, where Ω is a bounded and smooth domain. The operator A_Ω^s stands for two types of nonlocal operators that we treat in a unified way: either the spectral fractional Laplacian or the restricted fractional Laplacian. In both cases s ∈ (0,1) and the Dirichlet conditions are different: for the spectral fractional Laplacian, we prescribe u=0 on ∂ Ω and for the restricted fractional Laplacian, we prescribe u=0 on \mathbb Rⁿ ∖ Ω. We construct solutions when the exponent p = (n+2s)/(n-2s) ± ε is close to the critical one, concentrating as ε → 0 near critical points of a reduced function involving the Green and Robin functions of the domain

Citations

• Nonlinear equations for fractional Laplacians I: Regularity, maximum\n principles, and Hamiltonian estimates

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