Multiple sign-changing and semi-nodal normalized solutions for a Gross-Pitaevskii type system on bounded domain: the $L^2$-supercritical case
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908425512812544 |
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| author | Liu, Tianhao Song, Linjie Wu, Qiaoran Zou, Wenming |
| author_facet | Liu, Tianhao Song, Linjie Wu, Qiaoran Zou, Wenming |
| contents | In this paper we investigate the existence of multiple sign-changing and semi-nodal normalized solutions for an $m$-coupled elliptic system of the Gross-Pitaevskii type:
\begin{equation}
\left\{
\begin{aligned}
&-Δu_j + λ_j u_j = \sum_{k=1 }^mβ_{kj} u_k^2 u_j, \quad u_j \in H_0^1(Ω),
&\int_Ωu_j^2dx = c_j, \quad j = 1,2,\cdots,m.
\end{aligned}
\right.
\end{equation}
Here, $Ω\subset \mathbb{R}^N$ ($N = 3,4$) is a bounded domain. The constants $β_{kj} \neq 0$ and $c_j > 0$ are prescribed constants, while $λ_1, \cdots, λ_m$ are unknown and appear as Lagrange multipliers. This is the first result in the literature on the existence and multiplicity of sign-changing and semi-nodal normalized solutions of couple Schrödinger system in all regimes of $β_{kj}$. The main tool which we use is a new skill of vector linking and this article attempts for the first time to use linking method to search for solutions of a coupled system. Particularly, to obtain semi-nodal normalized solutions, we introduce partial vector linking which is new up to our knowledge. Moreover, by investigating the limit process as $\vec{c}=(c_1,\ldots,c_m) \to \vec{0}$ we obtain some bifurcation results. Note that when $N=4$, the system is of Sobolev critical. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_22152 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multiple sign-changing and semi-nodal normalized solutions for a Gross-Pitaevskii type system on bounded domain: the $L^2$-supercritical case Liu, Tianhao Song, Linjie Wu, Qiaoran Zou, Wenming Analysis of PDEs 35J50, 35J15, 35J60 In this paper we investigate the existence of multiple sign-changing and semi-nodal normalized solutions for an $m$-coupled elliptic system of the Gross-Pitaevskii type: \begin{equation} \left\{ \begin{aligned} &-Δu_j + λ_j u_j = \sum_{k=1 }^mβ_{kj} u_k^2 u_j, \quad u_j \in H_0^1(Ω), &\int_Ωu_j^2dx = c_j, \quad j = 1,2,\cdots,m. \end{aligned} \right. \end{equation} Here, $Ω\subset \mathbb{R}^N$ ($N = 3,4$) is a bounded domain. The constants $β_{kj} \neq 0$ and $c_j > 0$ are prescribed constants, while $λ_1, \cdots, λ_m$ are unknown and appear as Lagrange multipliers. This is the first result in the literature on the existence and multiplicity of sign-changing and semi-nodal normalized solutions of couple Schrödinger system in all regimes of $β_{kj}$. The main tool which we use is a new skill of vector linking and this article attempts for the first time to use linking method to search for solutions of a coupled system. Particularly, to obtain semi-nodal normalized solutions, we introduce partial vector linking which is new up to our knowledge. Moreover, by investigating the limit process as $\vec{c}=(c_1,\ldots,c_m) \to \vec{0}$ we obtain some bifurcation results. Note that when $N=4$, the system is of Sobolev critical. |
| title | Multiple sign-changing and semi-nodal normalized solutions for a Gross-Pitaevskii type system on bounded domain: the $L^2$-supercritical case |
| topic | Analysis of PDEs 35J50, 35J15, 35J60 |
| url | https://arxiv.org/abs/2506.22152 |