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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2406.13204 |
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Table of Contents:
- 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 $μ_1,μ_2,a_1,a_2,β>0$, $κ\in\mathbb{R}$, $V_1(x)$ and $V_2(x)$ are trapping potentials, and $λ_1,λ_2$ 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 $κ=0$ and $κ\neq 0$ respectively.