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Bubbling solutions for nonlocal elliptic problems
- Publication Year :
- 2014
-
Abstract
- We investigate bubbling solutions for the nonlocal equation \[ A_\Omega^s u =u^p,\ u >0 \quad \mbox{in } \Omega, \] under homogeneous Dirichlet conditions, where $\Omega$ is a bounded and smooth domain. The operator $A_\Omega^s$ 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 $s \in (0,1)$ 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 $\mathbb R^n \setminus \Omega$. We construct solutions when the exponent $p = (n+2s)/(n-2s) \pm \varepsilon$ is close to the critical one, concentrating as $\varepsilon \to 0$ near critical points of a reduced function involving the Green and Robin functions of the domain
- Subjects :
- Mathematics - Analysis of PDEs
35J91, 35B33, 35B20, 35B38
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.1410.5461
- Document Type :
- Working Paper