# Kirchhoff equation

## On the singularly perturbation fractional Kirchhoff equations: critical case

This paper deals with the following fractional Kirchhoff problem with critical exponent \begin{equation*} \Big(a+b{\int_{\mathbb{R}^{N}}}|(-\Delta)^{\frac{s}{2}}u|^2dx\Big)(-\Delta)^su=(1+\varepsilon K(x))u^{2^*_s-1}, \end{equation*} where $a,b>0$ are given constants, $\varepsilon$ is a small parameter, $2^*_s=\frac{2N}{N-2s}$ with $0<s<1$ and $N\geq4s$.

## Non-degeneracy of positive solutions for fractional Kirchhoff problems: high dimensional cases

In this paper, we establish the nondegeneracy of positive solutions to the fractional Kirchhoff problem \begin{equation*} \Big(a+b{\int_{\mathbb{R}^{N}}}|(-\Delta)^{\frac{s}{2}}u|^2dx\Big)(-\Delta)^su+u=u^p, \end{equation*} where $a,b>0$, $0<s<1$, $1<p<\frac{N+2s}{N-2s}$ and $(-\Delta )^s$ is the fractional Laplacian. In particular, we prove that uniqueness breaks down for dimensions $N>4s$, i.