Local uniqueness of semiclassical bounded states for a singularly perturbed fractional Kirchhoff problem

Abstract

In this paper, we consider the following singularly perturbed fractional Kirchhoff 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-2}u,\quad \text{in}\ \mathbb{R}^{N}, \end{equation*} where $a,b>0$, $2s<N<4s$ with $s\in(0,1)$, $2<p<2^*_s=\frac{2N}{N-2s}$ and $(-\Delta )^s$ is the fractional Laplacian. For $\varepsilon> 0$ sufficiently small and a bounded continuous function $V$, we establish a type of local Pohoz\v{a}ev identity by extension technique and then we can obtain the local uniqueness of semiclassical bounded solutions based on our recent results on the uniqueness and non-degeneracy of positive solutions to the limit problem.