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}}}|(-Δ)^{\frac{s}{2}}u|^2dx\Big)(-Δ)^su+V(x)u=|u|^{p-2}u,\quad \text{in}\ \mathbb{R}^{N}, \end{equation*} where $a,b>0$, $2s<N<4s$ with $s\in (0,1)$, $2<p<2_s^{\ast }=\frac{2N}{N-2s}$ and $(-\Delta {)}^s$ is the fractional Laplacian. For $\epsilon >0$ sufficiently small and a bounded continuous function $V$, we establish a type of local Pohozǎev 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.

28 pages, comments are welcome