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

For s ∈ ( 0 , 1 ) , N > 2 s , and a bounded open set Ω ⊂ R N with C 2 boundary, we study the fractional Brezis–Nirenberg type minimization problem of finding S ( a ) : = inf ∫ R N | ( - Δ ) s / 2 u | 2 + ∫ Ω a u 2 ∫ Ω u 2 N N - 2 s N - 2 s N , where the infimum is taken over all functions u ∈ H s ( 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 ∈ ( 2 s , 4 s ) , we prove that the Robin function ϕ a satisfies inf x ∈ Ω ϕ a ( x ) = 0 , which extends a result obtained by Druet for s = 1 . In dimensions N ∈ ( 8 s / 3 , 4 s ) , we then study the asymptotics of the fractional Brezis–Nirenberg energy S ( a + ε V ) 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.