For s∈ (0 , 1) , N> 2 s, and a bounded open set Ω ⊂ RN with C2 boundary, we study the fractional Brezis–Nirenberg type minimization problem of finding S(a):=inf∫RN|(-Δ)s/2u|2+∫Ωau2(∫Ωu2NN-2s)N-2sN,where the infimum is taken over all functions u∈ Hs(RN) that vanish outside Ω. 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 inf x∈Ωϕa(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+ 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.
Critical functions and blow-up asymptotics for the fractional Brezis–Nirenberg problem in low dimension
De Nitti N;
2023-01-01
Abstract
For s∈ (0 , 1) , N> 2 s, and a bounded open set Ω ⊂ RN with C2 boundary, we study the fractional Brezis–Nirenberg type minimization problem of finding S(a):=inf∫RN|(-Δ)s/2u|2+∫Ωau2(∫Ωu2NN-2s)N-2sN,where the infimum is taken over all functions u∈ Hs(RN) that vanish outside Ω. 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 inf x∈Ωϕa(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+ 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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
