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

Abstract

AbstractFor $$s \in (0,1)$$ s ∈ ( 0 , 1 ) , $$N > 2s$$ N > 2 s , and a bounded open set $$\Omega \subset {\mathbb {R}}^N$$ Ω ⊂ R N with $$C^2$$ C 2 boundary, we study the fractional Brezis–Nirenberg type minimization problem of finding $$\begin{aligned} S(a):= \inf \frac{\int _{{\mathbb {R}}^N} |(-\Delta )^{s/2} u|^2 + \int _\Omega a u^2}{\left( \int _\Omega u^\frac{2N}{N-2s} \right) ^\frac{N-2s}{N}}, \end{aligned}$$ 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 $$u \in H^s({\mathbb {R}}^N)$$ u ∈ H s ( R N ) that vanish outside $$\Omega $$ Ω . The function a is assumed to be critical in the sense of Hebey and Vaugon. For low dimensions $$N \in (2\,s, 4\,s)$$ N ∈ ( 2 s , 4 s ) , we prove that the Robin function $$\phi _a$$ ϕ a satisfies $$\inf _{x \in \Omega } \phi _a(x) = 0$$ inf x ∈ Ω ϕ a ( x ) = 0 , which extends a result obtained by Druet for $$s = 1$$ s = 1 . In dimensions $$N \in (8s/3, 4s)$$ N ∈ ( 8 s / 3 , 4 s ) , we then study the asymptotics of the fractional Brezis–Nirenberg energy $$S(a + \varepsilon V)$$ S ( a + ε V ) for some $$V \in L^\infty (\Omega )$$ V ∈ L ∞ ( Ω ) as $$\varepsilon \rightarrow 0+$$ 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.