A singularly perturbed fractional Kirchhoff problem

Abstract

In this paper, we first establish the uniqueness and non-degeneracy of positive solutions to the fractional Kirchhoff problem \begin{equation*} \Big(a+b{\int_{\mathbb{R}^{N}}}|(-Δ)^{\frac{s}{2}}u|^2dx\Big)(-Δ)^su+mu=|u|^{p-2}u,\quad \text{in}\ \mathbb{R}^{N}, \end{equation*} where $a,b,m>0$, $0<\frac{N}{4}<s<1$, $2<p<2_s^{\ast }=\frac{2N}{N-2s}$ and $(-\Delta {)}^s$ is the fractional Laplacian. Then, combining this non-degeneracy result and Lyapunov-Schmidt reduction method, we derive the existence of semiclassical solutions to the singularly perturbation 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*} for $\epsilon >0$ sufficiently small and a potential function $V$.

23 pages, comments are welcome