Symmetry and symmetry breaking for the fractional Caffarelli-Kohn-Nirenberg inequality

In this paper, we will consider the fractional Caffarelli-Kohn-Nirenb erg inequality & nbsp;lambda(integral(Rn)& nbsp;|u(x)|(p)/xI(beta P) dx)(2/p) <= integral(Rn)integral(Rn) (u(x) - u(y))(2)/ |x - y|(n+2 gamma)|x|(alpha)|y|(alpha) dy dx & nbsp;where gamma is an element of (0, 1), n & nbsp;>= 2, and alpha, beta & nbsp;is an element of R satisfy & nbsp;alpha & nbsp;<= beta & nbsp;<= alpha + gamma, -2 gamma < alpha < n - 2 gamma/2,& nbsp;and the exponent p is chosen to be & nbsp;p = 2n/n- 2 gamma + 2(beta - alpha),& nbsp;such that the inequality is invariant under scaling. We first study the existence and nonexistence of extremal solutions. Our next goal is to show some results on the symmetry and symmetry breaking region for the minimizers; these suggest the existence of a Felli-Schneider type curve separating both regions but, surprisingly, we find a novel behavior as alpha & nbsp;-> -2 gamma. The main idea in the proofs, as in the classical case, is to reformulate the fractional Caffarelli-Kohn-Nirenb erg inequality in cylindrical variables. Then, in order to find the radially symmetric solutions we need to solve a non-local ODE.& nbsp;For this equation we also get uniqueness of minimizers in the radial symmetry class; indeed, we show that the unique continuation argument of Frank-Lenzmann (Acta'13) can be applied to more general operators with good spectral properties. We provide, in addition, a completely new proof of non-degeneracy which works for all critical points. It is based on the variation of constants approach and the non local Wronskian of Ao-Chan-DelaTorre-Fontelos-Gonzalez-Wei (Duke'19). (C) 2022 Elsevier Inc. All rights reserved.