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©\times $\mathbb{R}$$forany$c>0$,$a,b >0$and$\frac{14}{3}\leq p< 6,$where$S©={u\in H^{1}($\mathbb{R}$^{3}):\int_{$\mathbb{R}$^{3}}u^{2}=c^{2}}.$ $\mathbf{\text{Keywords:}}$ Kirchhoff type equation; Critical nonlinearity; Normalized ground states \noindent{AMS Subject Classification:, 37L05; 35B40; 35B41.}