Bubbling solutions for nonlocal elliptic problems

TL;DR

This paper studies bubbling solutions for a class of nonlocal elliptic equations involving fractional Laplacians, constructing solutions that concentrate near critical points as parameters approach critical exponents.

Contribution

It provides a unified approach to analyze bubbling solutions for both spectral and restricted fractional Laplacians near critical exponents.

Findings

Constructed solutions concentrating near critical points

Unified treatment of spectral and restricted fractional Laplacians

Analyzed behavior as the exponent approaches the critical value

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