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

In this paper we study the nonlinear Bose–Einstein condensates Schrödinger system $\{\begin{array}{c}-\mathrm{\Delta }{u}_1-{\lambda }_1{u}_1={\mu }_1{u}_1^3+\beta u_1{u}_2^2+\kappa (x)u_2\text{in }{\mathbb{R}}^3,\hfill \\ -\mathrm{\Delta }{u}_2-{\lambda }_2{u}_2={\mu }_2{u}_2^3+\beta u_1^2{u}_2+\kappa (x)u_1\text{in }{\mathbb{R}}^3,\hfill \\ {\int }_{{\mathbb{R}}^3}{u}_1^2=a_1^2,{\int }_{{\mathbb{R}}^3}{u}_2^2=a_2^2,\hfill \end{array}$ \( \textstyle\begin{cases} -\Delta u_{1}-\lambda _{1} u_{1}=\mu _{1} u_{1}^{3}+\beta u_{1}u_{2}^{2}+ \kappa (x) u_{2}\quad\text{in }\mathbb{R}^{3}, \\ -\Delta u_{2}-\lambda _{2} u_{2}=\mu _{2} u_{2}^{3}+\beta u_{1}^{2}u_{2}+ \kappa (x) u_{1}\quad\text{in }\mathbb{R}^{3}, \\ \int _{\mathbb{R}^{3}} u_{1}^{2}=a_{1}^{2},\qquad \int _{\mathbb{R}^{3}} u_{2}^{2}=a_{2}^{2}, \end{cases} \) where $a_1$ $a_{1}$ , $a_2$ $a_{2}$ , ${\mu }_1$ $\mu _{1}$ , ${\mu }_2$ $\mu _{2}$ , $\kappa =\kappa (x)>0$ $\kappa =\kappa (x)>0$ , $\beta <0$ $\beta <0$ , and ${\lambda }_1$ $\lambda _{1}$ , ${\lambda }_2$ $\lambda _{2}$ 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.