Type

Publication

J. Geom. Anal. 32 (2022), no. 4, Paper No. 139, 24 pp

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.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. As one application, combining this nondegeneracy result and Lyapunov-Schmidt reduction method, we can derive the existence of solutions to the singularly perturbation problems.