On the singularly perturbation fractional Kirchhoff equations: critical case

Adv. Nonlinear Anal. 11 (2022), no. 1, 1097–1116

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 $N\geq4s$. We first prove the nondegeneracy of positive solutions when $\varepsilon=0$. In particular, we prove that uniqueness breaks down for dimensions $N>4s$, i.e., we show that there exist two non-degenerate positive solutions which seem to be completely different from the result of the fractional Schr"{o}dinger equation or the low dimensional fractional Kirchhoff equation. Using the finite-dimensional reduction method and perturbed arguments, we also obtain the existence of positive solutions to the singularly perturbation problems for $\varepsilon$ small.