Normalized vector solutions of nonlinear Schrödinger systems
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866910994137088000 |
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| author | Huang, Xiaomeng Pistoia, Angela Troestler, Christophe Wang, Chunhua |
| author_facet | Huang, Xiaomeng Pistoia, Angela Troestler, Christophe Wang, Chunhua |
| contents | Given $μ>0$ we look for solutions $ λ\in\mathbb{R}$ and $v_1,\dots,v_k\in H^1(\mathbb{R}^N)$ of the system \[ \begin{cases}
\displaystyle
-Δv_i+ λv_i+V_i(x)v_i
= \sum_{\substack{j=1}}^kβ_{ij} v_iv_j^2
&\text{ in } \mathbb{R}^N, \text{ } i=1,\dots,k,\newline
\displaystyle
\int_{\mathbb{R}^N} \left(v_1^2+\dots+v_k^2 \right)\mathrm{d} x = μ,
\end{cases}\] where $N=1,2,3$, $V_i:\mathbb R^N\to \mathbb R$ and $β_{ij}\in\mathbb{R}$ satisfy $β_{ij}=β_{ji}$ and $β_{ii}>0$. Under suitable assumptions on the $β_{ij}$'s, given a non-degenerate critical point $ξ_0$ of a suitable linear combination of the potentials $V_i$, we build solutions whose components concentrate at $ξ_0$ as the prescribed global mass $μ$ is either large (when $N=1$) or small (when $N=3$) or it approaches some critical threshold (when $N=2$). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_21940 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Normalized vector solutions of nonlinear Schrödinger systems Huang, Xiaomeng Pistoia, Angela Troestler, Christophe Wang, Chunhua Analysis of PDEs 35J47, 35B25, 35Q55 Given $μ>0$ we look for solutions $ λ\in\mathbb{R}$ and $v_1,\dots,v_k\in H^1(\mathbb{R}^N)$ of the system \[ \begin{cases} \displaystyle -Δv_i+ λv_i+V_i(x)v_i = \sum_{\substack{j=1}}^kβ_{ij} v_iv_j^2 &\text{ in } \mathbb{R}^N, \text{ } i=1,\dots,k,\newline \displaystyle \int_{\mathbb{R}^N} \left(v_1^2+\dots+v_k^2 \right)\mathrm{d} x = μ, \end{cases}\] where $N=1,2,3$, $V_i:\mathbb R^N\to \mathbb R$ and $β_{ij}\in\mathbb{R}$ satisfy $β_{ij}=β_{ji}$ and $β_{ii}>0$. Under suitable assumptions on the $β_{ij}$'s, given a non-degenerate critical point $ξ_0$ of a suitable linear combination of the potentials $V_i$, we build solutions whose components concentrate at $ξ_0$ as the prescribed global mass $μ$ is either large (when $N=1$) or small (when $N=3$) or it approaches some critical threshold (when $N=2$). |
| title | Normalized vector solutions of nonlinear Schrödinger systems |
| topic | Analysis of PDEs 35J47, 35B25, 35Q55 |
| url | https://arxiv.org/abs/2503.21940 |