Existence of normalized solutions for Schrödinger systems with linear and nonlinear couplings

In this paper we study the nonlinear Bose–Einstein condensates Schrödinger system $$ \textstyle\begin{cases} -\Delta u_{1}-\lambda _{1} u_{1}=\mu _{1} u_{1}^{3}+\beta u_{1}u_{2}^{2}+ \kappa (x) u_{2}\quad\text{in }\mathbb{R}^{3}, \\ -\Delta u_{2}-\lambda _{2} u_{2}=\mu _{2} u_{2}^{3}+\beta u_{1}^{2}u_{2}+ \kappa (x) u_{1}\quad\text{in }\mathbb{R}^{3}, \\ \int _{\mathbb{R}^{3}} u_{1}^{2}=a_{1}^{2},\qquad \int _{\mathbb{R}^{3}} u_{2}^{2}=a_{2}^{2}, \end{cases} $$ where $a_{1}$ , $a_{2}$ , $\mu _{1}$ , $\mu _{2}$ , $\kappa =\kappa (x)>0$ , $\beta <0$ , and $\lambda _{1}$ , $\lambda _{2}$ are Lagrangian multipliers. We use the Ekeland variational principle and the minimax method on manifold to prove that this system has a solution that is radially symmetric and positive.