Normalized ground states for Kirchhoff equations in ${\mathbb{R}}^{3}$ with a critical nonlinearity

This paper is concerned with the existence of ground states for a class of Kirchhoff type equation with combined power nonlinearities \begin{equation*} -\left(a+b\int_{\mathbb{R}^{3}}|\nabla u(x)|^{2}\right) \Delta u =\lambda u+|u|^{p-2}u+u^{5}\quad \ \text{for some} \ \lambda\in\mathbb{R},\quad x\in\mathbb{R}^{3}, \end{equation*} with prescribed $L^{2}$-norm mass \begin{equation*} \int_{\mathbb{R}^{3}}u^{2}=c^{2} \end{equation*} in Sobolev critical case and proves that the equation has a couple of solutions $(u_{c},\lambda_{c})\in S(c)\times \mathbb{R}$ for any $c>0$, $a,b >0$ and $\frac{14}{3}\leq p< 6,$ where $S(c)=\{u\in H^{1}(\mathbb{R}^{3}):\int_{\mathbb{R}^{3}}u^{2}=c^{2}\}.$ \textbf{Keywords:} Kirchhoff type equation; Critical nonlinearity; Normalized ground states \noindent{AMS Subject Classification:\, 37L05; 35B40; 35B41.}