Bubbling solutions for nonlocal elliptic problems

• Juan Dávila ^[1] ; Luis López Ríos ^[2] ; Yannick Sire ^[3]
  1. [1] Universidad de Chile

     Universidad de Chile

     Santiago, Chile

     
  2. [2] Universidad de Buenos Aires

     Universidad de Buenos Aires

     Argentina

     
  3. [3] Université d' Aix-Marseille

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• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 33, Nº 2, 2017, págs. 509-546
• Idioma: inglés
• DOI: 10.4171/RMI/947
• Texto completo no disponible (Saber más ...)
• Resumen

  • We investigate bubbling solutions for the nonlocal equation AsΩu=up, u>0in Ω, under homogeneous Dirichlet conditions, where Ω is a bounded and smooth domain. The operator AsΩ 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 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.