Normalized solutions to Schr\"{o}dinger systems with potentials

In this paper, we study the normalized solutions of the Schr\"{o}dinger system with trapping potentials {equation}{eq:diricichlet} {cases} -Δ u_1+V_1(x)u_1-λ_1 u_1=μ_1 u_1^3+β u_1u_2²+κ u_2~in~ R^3,\\ -Δ u_2+V_2(x)u_2-λ_2 u_2=μ_2 u_2^3+β u_1^2u_2+κ u_1~in~ R^3, u_1∈ H^1(R^3), u_2∈ H^1(R^3), {cases} {equation} under the constraint {equation} ∫R^3 u_1^2=a_1^2,~∫R^3 u_2^2=a_2^2, {equation} where μ₁,μ₂,a₁,a₂,β>0, κ, V₁(x) and V₂(x) are trapping potentials, and λ₁,λ₂ are lagrangian multipliers, this is a typical L²-supercritical case in R³. We obtain the existence of solutions to this system by minimax theory on the manifold for κ=0 and κ≠ 0 respectively.