Normalized solution for a kind of coupled Kirchhoff systems

In this paper, we investigate the existence of a normalized solution for the following Kirchhoff system in the entire space ${\mathbb{R}}^N$ ( $N\ge 3$):

$$\begin{array}{r}\{\begin{array}{l}-\left(1+{\int }_{{\mathbb{R}}^N}|\mathrm{\nabla }u{|}^{2}dx\right)\mathrm{\Delta }u={\lambda }_{1}u+{\mu }_1|u{|}^{p-2}u+\beta r_1|u{|}^{r_1-2}u|v{|}^{r_2},\\ -\left(1+{\int }_{{\mathbb{R}}^N}|\mathrm{\nabla }v{|}^{2}dx\right)\mathrm{\Delta }v={\lambda }_{2}v+{\mu }_2|v{|}^{q-2}v+\beta r_2|u{|}^{r_1}|v{|}^{r_2-2}v,\end{array}(\mathrm{P})\end{array}$$

under the constraints ${\int }_{{\mathbb{R}}^N}|u{|}^{2}dx=m_1\text{and}{\int }_{{\mathbb{R}}^N}|v{|}^{2}dx=m_2$, where $m_1,m_2>0$ are prescribed. The parameters ${\mu }_1,{\mu }_2,\beta >0$, $2\le p,q<2+\frac{8}{N}$, $r_1,r_2>1,$ and satisfy $r_1+r_2=2^{\ast }=\frac{2N}{N-2}$. The frequencies ${\lambda }_1,{\lambda }_2$ appear as Lagrange multipliers. With the help of the Pohožaev manifold and the minimization of the energy functional over a combination of the mass constraints and the closed balls, we obtain a positive ground state solution to (P). We mainly extend the results of Yang (Normalized ground state solutions for Kirchhoff-type systems) concerning the above problem from a single critical to a coupled critical nonlinearity.