Existence and multiplicity of $L^2$-Normalized solutions for the periodic Schrödinger system of Hamiltonian type
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arXiv
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2025
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| author | Qiu, Ruowen Yu, Yuanyang Zhao, Fukun |
| author_facet | Qiu, Ruowen Yu, Yuanyang Zhao, Fukun |
| contents | In this paper, we study the following nonlinear Schrödinger system of Hamiltonian type \begin{equation*} \left\{\begin{array}{l} -Δu+V(x)u=\partial_v H(x,u,v)+ωv, \ x \in \mathbb{R}^N, \\ -Δv+V(x)v=\partial_u H(x,u,v)+ωu,\ x \in \mathbb{R}^N, \\ \displaystyle\int_{\mathbb{R}^N}|z|^2dx=a^2, \end{array}\right. \end{equation*} where the potential function $V(x)$ is periodic, $z:=(u,v):\mathbb{R}^N\rightarrow \mathbb{R}\times\mathbb{R}$, $ω\in \mathbb{R}$ appears as a Lagrange multiplier, $a>0$ is a prescribed constant. The existence and multiplicity of $L^2$-normalized solutions for the above Schrödinger system are obtained, and the combination of the Lyapunov-Schmidt reduction, a perturbation argument and the multiplicity theorem of Ljusternik-Schnirelmann is involved in the proof. In addition, a bifurcation result is also given. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_15656 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Existence and multiplicity of $L^2$-Normalized solutions for the periodic Schrödinger system of Hamiltonian type Qiu, Ruowen Yu, Yuanyang Zhao, Fukun Analysis of PDEs 35J50(Primary), 35J47(Secondary), 35B40 In this paper, we study the following nonlinear Schrödinger system of Hamiltonian type \begin{equation*} \left\{\begin{array}{l} -Δu+V(x)u=\partial_v H(x,u,v)+ωv, \ x \in \mathbb{R}^N, \\ -Δv+V(x)v=\partial_u H(x,u,v)+ωu,\ x \in \mathbb{R}^N, \\ \displaystyle\int_{\mathbb{R}^N}|z|^2dx=a^2, \end{array}\right. \end{equation*} where the potential function $V(x)$ is periodic, $z:=(u,v):\mathbb{R}^N\rightarrow \mathbb{R}\times\mathbb{R}$, $ω\in \mathbb{R}$ appears as a Lagrange multiplier, $a>0$ is a prescribed constant. The existence and multiplicity of $L^2$-normalized solutions for the above Schrödinger system are obtained, and the combination of the Lyapunov-Schmidt reduction, a perturbation argument and the multiplicity theorem of Ljusternik-Schnirelmann is involved in the proof. In addition, a bifurcation result is also given. |
| title | Existence and multiplicity of $L^2$-Normalized solutions for the periodic Schrödinger system of Hamiltonian type |
| topic | Analysis of PDEs 35J50(Primary), 35J47(Secondary), 35B40 |
| url | https://arxiv.org/abs/2504.15656 |