Positive solutions for the fractional Kirchhoff type problem in exterior domains

In this article, we consider the following Kirchhoff equation involving fractional Laplacian \(\begin{aligned} \left\{ \begin{array}{ll} \left( a+b[u]_s^2\right) (-\Delta )^s u+u=|u|^{p-2}u &{}\text{ in }\ {\Omega }, \\ u=0&{}\text{ on }\ {{\mathbb {R}}^3 \backslash \Omega }, \\ \end{array} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}\left(a,+,b,{[u]}_s^2\right){(-\mathrm{\Delta })}^{s}u+u={|u|}^{p-2}u\hfill & \text{in}\mathrm{\Omega },\hfill \\ u=0\hfill & \text{on}{\mathbb{R}}^3\backslash \mathrm{\Omega },\hfill \end{array}\right)\end{array}$$ where \(a,\ b>0\) $a,b>0$ are constants, \(\frac{3}{4}<s<1,\) $\frac{3}{4}<s<1,$ \([u]_s\) ${[u]}_s$ is the so-called Gagliardo (semi)norm of u, \(4<p<2^*_{s}=\frac{6}{3-2s}\) $4<p<2_s^{\ast }=\frac{6}{3-2s}$ and \(\Omega \subset {\mathbb {R}}^3\) $\mathrm{\Omega }\subset {\mathbb{R}}^3$ is an exterior domain with smooth boundary \(\partial \Omega \ne \emptyset .\) $\partial \mathrm{\Omega }\ne \mathrm{\varnothing }.$ By establishing a global compactness lemma of the fractional Kirchhoff equation in exterior domains, we verify the compactness of Palais–Smale sequences corresponding to above problem at higher energy level interval. Then combining some crucial estimates and barycentric function, we determine the existence of positive bound state solutions provided that \({\mathbb {R}}^3\backslash \Omega \) ${\mathbb{R}}^3\backslash \mathrm{\Omega }$ is contained in a small ball. In addition, we point out that the main result can be extended to fractional Sobolev critical case with small parameter.