Normalized solutions to Schrödinger systems with potentials

In this paper, we study the normalized solutions of the Schrödinger system with trapping potentials \begin{equation}\label{eq:diricichlet} \begin{cases} -Δu_1+V_1(x)u_1-λ_1 u_1=μ_1 u_1^3+βu_1u_2^{2}+κu_2~\hbox{in}~ $\mathbb{R}$^3,\ -Δu_2+V_2(x)u_2-λ_2 u_2=μ_2 u_2^3+βu_1^2u_2+κu_1~\hbox{in}~ $\mathbb{R}$^3, u_1\in H^1($\mathbb{R}$^3), u_2\in H^1($\mathbb{R}$^3),\nonumber \end{cases} \end{equation} under the constraint \begin{equation} \int_{$\mathbb{R}$^3} u_1^2=a_1^2,~\int_{$\mathbb{R}$^3} u_2^2=a_2^2\nonumber, \end{equation} where ${\mu }_1,{\mu }_2,a_1,a_2,\beta >0$, $\kappa \in \mathbb{R}$, $V_1(x)$ and $V_2(x)$ are trapping potentials, and ${\lambda }_1,{\lambda }_2$ are lagrangian multipliers, this is a typical $L^2$-supercritical case in ${\mathbb{R}}^3$. We obtain the existence of solutions to this system by minimax theory on the manifold for $\kappa =0$ and $\kappa \mathrm{\ne }0$ respectively.