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

TL;DR

This paper proves the local uniqueness of semiclassical bounded solutions for a singularly perturbed fractional Kirchhoff problem using a new local Pohozhaev identity and recent results on the limit problem.

Contribution

It introduces a local Pohozhaev identity for the fractional Kirchhoff problem and establishes local uniqueness of solutions based on recent limit problem results.

Findings

Established a local Pohozhaev identity for the problem.

Proved local uniqueness of semiclassical solutions.

Extended recent results on the limit problem to the Kirchhoff setting.

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.