- journal article metadata
European Mathematical Society Publishing House
2017-05-12 23:45:01
Revista Matemática Iberoamericana
Rev. Mat. Iberoamericana
RMI
0213-2230
2235-0616
General
10.4171/RMI
http://www.ems-ph.org/doi/10.4171/RMI
subscribers
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2012)
33
2017
2
Bubbling solutions for nonlocal elliptic problems
Juan
Dávila
Universidad de Chile, SANTIAGO, CHILE
Luis
López Ríos
Universidad de Buenos Aires, BUENOS AIRES, ARGENTINA
Yannick
Sire
Université Aix-Marseille, MARSEILLE CEDEX 13, FRANCE
Fractional Laplacian, Dirichlet problem, sub and supercritical exponents, stable critical points
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.
Partial differential equations
Global analysis, analysis on manifolds
509
546
10.4171/RMI/947
http://www.ems-ph.org/doi/10.4171/RMI/947