Normalized solutions to Schr\"{o}dinger systems with linear and nonlinear couplings

In this paper, we study important Schr"{o}dinger systems with linear and nonlinear couplings \begin{equation}\label{eq:diricichlet} \begin{cases} -\Delta u_1-\lambda_1 u_1=\mu_1 |u_1|^{p_1-2}u_1+r_1\beta |u_1|^{r_1-2}u_1|u_2|^{r_2}+\kappa (x)u_2~\hbox{in}~$\mathbb{R}$^N,\ -\Delta u_2-\lambda_2 u_2=\mu_2 |u_2|^{p_2-2}u_2+r_2\beta |u_1|^{r_1}|u_2|^{r_2-2}u_2+\kappa (x)u_1~ \hbox{in}~$\mathbb{R}$^N,\ u_1\in H^1($\mathbb{R}$^N), u_2\in H^1($\mathbb{R}$^N),\nonumber \end{cases} \end{equation} with the condition $${\int }_{{\mathbb{R}}^N}{u}_1^2=a_1^2,{\int }_{{\mathbb{R}}^N}{u}_2^2=a_2^2,$$ where $N\ge 2$, ${\mu }_1,{\mu }_2,a_1,a_2>0$, $\beta \in \mathbb{R}$, $2<p_1,p_2<2^{\ast }$, $2<r_1+r_2<2^{\ast }$, $\kappa(x)\in L^{\infty}($\mathbb{R}$^N)$withfixedsignand$\lambda_1,\lambda_2$ are Lagrangian multipliers. We use Ekland variational principle to prove this system has a normalized radially symmetric solution for $L^2-$subcritical case when $N\ge 2$, and use minimax method to prove this system has a normalized radially symmetric positive solution for $L^2-$supercritical case when $N=3$, $p_1=p_2=4,r_1=r_2=2$.