Local uniqueness of multi-peak positive solutions to a class of fractional Kirchhoff equations

Abstract: This paper is twofold. In the first part, combining the nondegeneracy result and Lyapunov-Schmidt reduction method, we derive the existence of multi-peak positive solutions to the singularly perturbation problem \begin{equation*} \Big(\varepsilon^{2s}a+\varepsilon^{4s-N} b{\int_{\mathbb{R}^{N}}}|(-\Delta)^{\frac{s}{2}}u|^2dx\Big)(-\Delta)^su+V(x)u=u^p,\quad \text{in}\ \mathbb{R}^{N}, \end{equation*} for $\varepsilon> 0$ sufficiently small, $2s<N<4s$, $1<p<2^*_s-1$ and some mild assumptions on the function $V$. The main difficulties are from the nonlocal operator mixed the nonlocal term, which cause the corresponding unperturbed problem turns out to be a system of partial differential equations, but not a single fractional Kirchhoff equation. In the second part, under some assumptions on $V$, we show the local uniqueness of positive multi-peak solutions by using the local Pohoz\v{a}ev identity.