Critical functions and blow-up asymptotics for the fractional Brezis--Nirenberg problem in low dimension

Abstract

For $s\in (0,1)$ and a bounded open set $\Omega \subset {\mathbb{R}}^N$ with $N>2s$, we study the fractional Brezis--Nirenberg type minimization problem of finding $$S(a):=\inf \frac{{\int }_{{\mathbb{R}}^N}|(-\Delta {)}^{s/2}u{|}^2+{\int }_{\Omega }a{u}^2}{{\left({\int }_{\Omega }{u}^{\frac{2N}{N-2s}}\right)}^{\frac{N-2s}{N}}},$$ where the infimum is taken over all functions $u\in H^s({\mathbb{R}}^N)$ that vanish outside $\Omega$. 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 ${\varphi }_a$ satisfies ${\inf }_{x\in \Omega }{\varphi }_a(x)=0$, 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 $S(a+\epsilon V)$ for some $V\in L^{\infty }(\Omega )$ as $\epsilon \to 0+$. 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.

References in corpus (1)

• Blow-up of solutions of critical elliptic equations in three dimensions

Cited by in corpus (3)

• On the sharp constant in the Bianchi-Egnell stability inequality
• Stability with explicit constants of the critical points of the fractional Sobolev inequality and applications to fast diffusion
• Subelliptic Nonlocal Brezis-Nirenberg Problems on Stratified Lie Groups