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 $\phi_a$ satisfies $\inf_{x \in \Omega} \phi_a(x) = 0$, 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 $S(a + \varepsilon V)$ for some $V \in L^\infty(\Omega)$ as $\varepsilon \to 0+$. 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.