Normalized ground state solutions of Schrödinger-KdV system in $\mathbb{R}^3$
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914946504196096 |
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| author | Gao, Qian Wang, Qun Chang, Xiaojun |
| author_facet | Gao, Qian Wang, Qun Chang, Xiaojun |
| contents | In this paper, we study the coupled Schrödinger-KdV system
\begin{align*} \begin{cases}
-Δu +λ_1 u=u^3+βuv~~&\text{in}~~\mathbb{R}^{3}, \\-Δv +λ_2 v=\frac{1}{2}v^2+\frac{1}{2}βu^2~~&\text{in}~~\mathbb{R}^{3} \end{cases} \end{align*} subject to the mass constraints \begin{equation*} \int_{\mathbb{R}^{3}}|u|^2 dx=a,\quad \int_{\mathbb{R}^{3}}|v|^2 dx=b, \end{equation*}
where $a, b>0$ are given constants, $β>0$, and the frequencies $λ_1,λ_2$ arise as Lagrange multipliers. The system exhibits $L^2$-supercritical growth. Using a novel constraint minimization approach, we demonstrate the existence of a local minimum solution to the system. Furthermore, we establish the existence of normalized ground state solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_06528 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Normalized ground state solutions of Schrödinger-KdV system in $\mathbb{R}^3$ Gao, Qian Wang, Qun Chang, Xiaojun Analysis of PDEs 35J20, 35J60, 35Q55 In this paper, we study the coupled Schrödinger-KdV system \begin{align*} \begin{cases} -Δu +λ_1 u=u^3+βuv~~&\text{in}~~\mathbb{R}^{3}, \\-Δv +λ_2 v=\frac{1}{2}v^2+\frac{1}{2}βu^2~~&\text{in}~~\mathbb{R}^{3} \end{cases} \end{align*} subject to the mass constraints \begin{equation*} \int_{\mathbb{R}^{3}}|u|^2 dx=a,\quad \int_{\mathbb{R}^{3}}|v|^2 dx=b, \end{equation*} where $a, b>0$ are given constants, $β>0$, and the frequencies $λ_1,λ_2$ arise as Lagrange multipliers. The system exhibits $L^2$-supercritical growth. Using a novel constraint minimization approach, we demonstrate the existence of a local minimum solution to the system. Furthermore, we establish the existence of normalized ground state solutions. |
| title | Normalized ground state solutions of Schrödinger-KdV system in $\mathbb{R}^3$ |
| topic | Analysis of PDEs 35J20, 35J60, 35Q55 |
| url | https://arxiv.org/abs/2409.06528 |