Bubbling solutions for nonlocal elliptic problems

Juan Dávila, Luis López Ríos, Yannick Sire

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Abstract

We investigate bubbling solutions for the nonlocal equationAsΩu=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.
Original languageEnglish
Pages (from-to)509-546
Number of pages38
JournalRevista Matemática Iberoamericana
Volume33
Issue number2
Early online date9 May 2017
DOIs
Publication statusPublished - 31 Dec 2017

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