Normalized solutions for coupled Kirchhoff equations with critical and subcritical nonlinearities

In this paper, we study Kirchhoff equations with constraint conditions \begin{equation}\tag{P}\label{00001} \left\{ \begin{aligned} -\bigg(&a+b \int_{\mathbb{R}^{3}}|\nabla u_{1}|^{2} \, d x\bigg)\,\Delta u_{1} = \lambda_{1} u_{1}\\ &+\mu_{1}|u_{1}|^{p_{1}-2} u_{1} +\beta r_{1}|u_{1}|^{r_{1}-2} u_{1}|u_{2}|^{r_{2}} \quad \text{in } \mathbb{R}^{3}, \\ -\bigg(&a+b \int_{\mathbb{R}^{3}}|\nabla u_{2}|^{2} \, d x\bigg)\, \Delta u_{2} = \lambda_{2} u_{2}\\ &+\mu_{2}|u_{2}|^{p_{2}-2} u_{2} +\beta r_{2}|u_{1}|^{r_{1}}|u_{2}|^{r_{2}-2} u_{2} \quad \text{in } \mathbb{R}^{3}, \\ \int_{\mathbb{R}^{3}} &|u_{1}|^{2}\,dx=c_{1},\quad \int_{\mathbb{R}^{3}} |u_{2}|^{2}\,dx=c_{2}, \\ u&_{1} \in H^{1}\left(\mathbb{R}^{3}\right),\quad u_{2} \in H^{1}\left(\mathbb{R}^{3}\right). \end{aligned} \right. \end{equation} where \(a\), \(b\), \(\beta\), \(\mu_{i}\), \(c_{i}>0\), \(r_{i}>1\), \(2<p_{i}<\frac{14}{3}<r:=r_{1}+r_{2}\leq2^{*}\) for \(i=1\), \(2\), and \(\lambda_{1}\), \(\lambda_{2}\in \mathbb{R}\) appear as Lagrange multipliers. The existence of normalized solutions for \(p_1\) and \( p_2\) within a specific range of \((2, \frac{14}{3})\) has been considered both the Sobolev subcritical case (\(r < 2^{*}\)) and the critical case (\(r = 2^{*}\)) by the Minimax principle and variational methods. This paper provides a refinement and extension of the results for the normalized solutions to Kirchhoff equations.