Symmetry breaking of extremals for the high-order Caffarelli–Kohn–Nirenberg-type inequalities

In this paper, we give the first result about the precise symmetry and symmetry breaking regions of extremal functions for weighted second-order inequalities. First, based on the work of C.-S. Lin [Interpolation inequalities with weights, Comm. Partial Differential Equations 11(4) (1986) 1515–1538], a new second-order Caffarelli–Kohn–Nirenberg-type inequality will be established, i.e. [Formula: see text] [Formula: see text] for some constant [Formula: see text], where [Formula: see text] We obtain a symmetry breaking conclusion: when [Formula: see text] and [Formula: see text] where [Formula: see text], then the extremal function for the best constant [Formula: see text], if it exists, is non-radial. Furthermore, we give a symmetry result when [Formula: see text] and [Formula: see text], i.e. for all [Formula: see text], [Formula: see text] [Formula: see text] where [Formula: see text] is the sharp constant of second-order Sobolev inequality, and equality holds only for some radial functions of the form [Formula: see text] (up to scalings and multiplications).