Critical functions and blow-up asymptotics for the fractional Brezis--Nirenberg problem in low dimension
Nicola De Nitti, Tobias König
TL;DR
This work analyzes the fractional Brezis–Nirenberg problem on bounded domains for low dimensions $2s<N<4s$ with a Hebey–Vaugon critical potential $a$. It extends Druet’s classical $s=1$ result to the fractional setting by proving the Robin function satisfies $\inf_{x\in\Omega} \phi_a(x)=0$ in this regime, linking criticality to the nonexistence of minimizers and to concentration phenomena. For $N\in(8s/3,4s)$ it derives sharp blow-up asymptotics for the energy $S(a+\varepsilon V)$ as $\varepsilon\to0^+$, providing a detailed description of the blow-up profile, concentration speed, and concentration points through a refined expansion that accounts for the nonzero critical function $a$. The analysis introduces a precise bubble decomposition with projected bubbles $PU_{x,\lambda}$, a two-tier coercivity framework, and a careful control of remainder terms, yielding a comprehensive picture of how the energy dips below the Sobolev constant and where minimizers concentrate. These results extend the fractional Brezis–Nirenberg theory to the critical-function setting and illuminate the delicate interplay between Green's function theory, Robin functions, and concentration-compactness in low dimensions with nonlocal operators.
Abstract
For $s \in (0,1)$ and a bounded open set $Ω\subset \mathbb R^N$ with $N > 2s$, we study the fractional Brezis--Nirenberg type minimization problem of finding $$ S(a) := \inf \frac{\int_{\mathbb R^N} |(-Δ)^{s/2} u|^2 + \int_Ωa u^2}{\left( \int_Ωu^\frac{2N}{N-2s} \right)^\frac{N-2s}{N}}, $$ where the infimum is taken over all functions $u \in H^s(\mathbb R^N)$ that vanish outside $Ω$. The function $a$ is assumed to be critical in the sense of Hebey and Vaugon. For low dimensions $N \in (2s, 4s)$, we prove that the Robin function $φ_a$ satisfies $\inf_{x \in Ω} φ_a(x) = 0$, which extends a result obtained by Druet for $s = 1$. In dimensions $N \in (8s/3, 4s)$, we then study the asymptotics of the fractional Brezis--Nirenberg energy $S(a + \varepsilon V)$ for some $V \in L^\infty(Ω)$ as $\varepsilon \to 0+$. We give a precise description of the blow-up profile of (almost) minimizing sequences and characterize the concentration speed and the location of concentration points.