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\geq3 $):

$ \begin{align*} \begin{cases}-\left(1+\int_{ \mathbb{R}^N}|\nabla u|^2dx\right)\Delta u = \lambda_1u+\mu_1|u|^{p-2}u+\beta r_1|u|^{r_1-2}u|v|^{r_2}, \\ -\left(1+\int_{ \mathbb{R}^N}|\nabla v|^2dx\right)\Delta v = \lambda_2v+\mu_2|v|^{q-2}v+\beta r_2|u|^{r_1}|v|^{r_2-2}v, \end{cases}\;\;\;\;\;\;\;\;\;\;\;\;({\rm{P}}) \end{align*} $

under the constraints $ \int_{ \mathbb{R}^N }|u|^2dx = m_1 \ \mbox{and} \ \int_{ \mathbb{R}^N }|v|^2dx = m_2 $, where $ m_1, m_2 > 0 $ are prescribed. The parameters $ \mu_1, \mu_2, \beta > 0 $, $ 2\leq p, q < 2+\frac{8}{N} $, $ r_1, r_2 > 1, $ and satisfy $ {r_1}+{r_2} = 2^* = \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.