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

Abstract

For s in (0,1), N > 2s, and a bounded open set Omega subset {mathbb {R}}^N with C^2 boundary, we study the fractional Brezis–Nirenberg type minimization problem of finding S(a):=inf∫RN|(-Δ)s/2u|2+∫Ωau2∫Ωu2NN-2sN-2sN,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} S(a):= \\inf \\frac{\\int _{{\\mathbb {R}}^N} |(-\\Delta )^{s/2} u|^2 + \\int _\\Omega a u^2}{\\left( \\int _\\Omega u^\\frac{2N}{N-2s} \\right) ^\\frac{N-2s}{N}}, \\end{aligned}$$\\end{document}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 (2,s, 4,s), 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 rightarrow 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.