Revista Matemática Iberoamericana

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Volume 33, Issue 2, 2017, pp. 509–546
DOI: 10.4171/RMI/947

Bubbling solutions for nonlocal elliptic problems

Juan Dávila[1], Luis López Ríos[2] and Yannick Sire[3]

(1) Departamento de Ingenieria Matematica, CMM (UMR CN, Universidad de Chile, Casilla 170/3 Correo 3, Santiago de Chile, Chile
(2) Departamento de Matemática, Universidad de Buenos Aires, Intendente Güiraldes 2160 - Ciudad Universitaria, 1428, Buenos Aires, Argentina
(3) Laboratoire d’Analyse, Topologie, Probabilités UMR 7353, Université Aix-Marseille, Technopôle Château-Gombert, 9, rue F. Joliot Curie, 13453, Marseille CEDEX 13, France

We investigate bubbling solutions for the nonlocal equation $$A^s_{\Omega} u =u^p,\ u > 0 \quad \mbox{in } \Omega,$$ under homogeneous Dirichlet conditions, where $\Omega$ is a bounded and smooth domain. The operator $A^s_{\Omega}$ 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 $s \in (0,1)$, and the Dirichlet conditions are different: for the spectral fractional Laplacian, we prescribe $u=0$ on $\partial \Omega$, and for the restricted fractional Laplacian, we prescribe $u=0$ on $\mathbb R^n \backslash \Omega$. We construct solutions when the exponent $p = (n+2s)/(n-2s) \pm \epsilon$ is close to the critical one, concentrating as $\epsilon \to 0$ near critical points of a reduced function involving the Green and Robin functions of the domain.

Keywords: Fractional Laplacian, Dirichlet problem, sub and supercritical exponents, stable critical points

Dávila Juan, López Ríos Luis, Sire Yannick: Bubbling solutions for nonlocal elliptic problems. Rev. Mat. Iberoamericana 33 (2017), 509-546. doi: 10.4171/RMI/947