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

Abstract

The function a is assumed to be critical in the sense of Hebey and Vaugon. For low dimensions N∈(2s,4s) , we prove that the Robin function ϕa satisfies infx∈Ωϕa(x)=0, which extends a result obtained by Druet for s=1 . In dimensions N∈(8s/3,4s), we then study the asymptotics of the fractional Brezis–Nirenberg energy S(a+εV) for some V∈L∞(Ω) as ε→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.