Normalized solutions to Kirchhoff type equations with a critical growth nonlinearity

In this paper, we are concerned with normalized solutions of the Kirchhoff type equation \begin{equation*} -M\left(\int_{\R^N}|\nabla u|^2$\mathrm{d}$ x\right)Δu = λu +f(u) \ \ $\mathrm{i}\mathrm{n}$ \ \ $\mathbb{R}$^N \end{equation*} with $u\in S_c:=\{u\in H^1({\mathbb{R}}^N):{\int }_{{\mathbb{R}}^N}{u}^2\mathrm{d}x=c^2\}$. When $N=2$ and $f$ has exponential critical growth at infinity, normalized mountain pass type solutions are obtained via the variational methods. When $N\ge 4$, $M(t)=a+bt$ with $a$, $b>0$ and $f$ has Sobolev critical growth at infinity, we investigate the existence of normalized ground state solutions and normalized mountain pass type solutions. Moreover, the non-existence of normalized solutions is also considered.