Normalized solutions for Kirchhoff type equations with combined nonlinearities: The Sobolev critical case

In this paper, we study the Kirchhoff equation with Sobolev critical exponent \begin{document}$ -\left(a+b\int_{ {\mathbb{R}}^3}|\nabla u|^2\right)\Delta u = \lambda u+\mu|u|^{q-2}u+|u|^{4}u\ \ {\rm in}\ {\mathbb{R}}^3 $\end{document} under the normalized constraint$ \int_{ {\mathbb{R}}^3}u^2 = c^2, $where \begin{document}$ a, \, b, \, c>0 $\end{document} are constants, \begin{document}$ \lambda, \, \mu\in{\mathbb{R}} $\end{document} and \begin{document}$ 2<q<6 $\end{document}. The number \begin{document}$ 2+8/3 $\end{document} behaves as the \begin{document}$ L^2 $\end{document}-critical exponent for the above problem. When \begin{document}$ \mu>0 $\end{document}, we distinguish the problem into four cases: \begin{document}$ 2<q<2+4/3 $\end{document}, \begin{document}$ q = 2+4/3 $\end{document}, \begin{document}$ 2+4/3<q<2+8/3 $\end{document} and \begin{document}$ 2+8/3\leq q<6 $\end{document}, and prove the existence and multiplicity of normalized solutions under suitable assumptions on \begin{document}$ \mu $\end{document} and \begin{document}$ c $\end{document}. The solution obtained is either a minimum (local or global) or a mountain pass solution. When \begin{document}$ \mu\leq 0 $\end{document}, we establish the nonexistence of nonnegative normalized solutions. Finally, we investigate the asymptotic behavior of normalized solutions obtained above as \begin{document}$ \mu\to0^+ $\end{document} and as \begin{document}$ b\to0^+ $\end{document} respectively.