Critical fractional Kirchhoff problems: Uniqueness and Nondegeneracy

In this paper, we consider the following critical fractional Kirchhoff equation \begin{equation*} \Big(a+b{\int_{$\mathbb{R}$^{N}}}|(-Δ)^{\frac{s}{2}}u|^2dx\Big)(-Δ)^su=|u|^{2^_s-2}u,\quad $\text{in}$\ $\mathbb{R}$^{N}, \end{equation} where $a,b>0$, $\frac{N}{4}<s<1$, $2_s^{\ast }=\frac{2N}{N-2s}$ and $(-\mathrm{\Delta }{)}^s$ is the fractional Laplacian. We prove the uniqueness and nondegeneracy of positive solutions to the problem, which can be used to study the singular perturbation problems concerning fractional Kirchhoff equations.