Ground state and nodal solutions for fractional Kirchhoff equation with pure critical growth nonlinearity

In this paper, we consider the existence of least energy nodal solution and ground state solution, energy doubling property for the following fractional critical problem\begin{document}$ \begin{cases} -(a+ b\|u\|_{K}^{2})\mathcal{L}_K u+V(x)u = |u|^{2^{\ast}_{\alpha}-2}u+ k f(x,u),&x\in\Omega,\\ u = 0,&x\in\mathbb{R}^{3}\backslash\Omega, \end{cases} $\end{document}where \begin{document}$ k $\end{document} is a positive parameter, \begin{document}$ \mathcal{L}_K $\end{document} stands for a nonlocal fractional operator which is defined with the kernel function \begin{document}$ K $\end{document} . By using the nodal Nehari manifold method, we obtain a least energy nodal solution \begin{document}$ u $\end{document} and a ground state solution \begin{document}$ v $\end{document} to this problem when \begin{document}$ k\gg1 $\end{document} , where the nonlinear function \begin{document}$ f:\mathbb{R}^{3}\times\mathbb{R}\rightarrow \mathbb{R} $\end{document} is a Carathéodory function.