A Liouville-type theorem for the coupled Schrödinger systems and the uniqueness of the sign-changing radial solutions
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| Format: | Preprint |
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2024
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| _version_ | 1866911767297261568 |
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| author | Li, Haoyu Miyagaki, Olímpio Hiroshi |
| author_facet | Li, Haoyu Miyagaki, Olímpio Hiroshi |
| contents | In this paper, we study the sign-changing radial solutions of the following coupled Schrödinger system \begin{equation}
\left\{
\begin{array}{lr}
-Δu_j+λ_j u_j=μ_j u_j^3+\sum_{i\neq j}β_{ij} u_i^2 u_j \,\,\,\,\,\,\,\, \mbox{in }B_1 ,\nonumber
u_j\in H_{0,r}^1(B_1)\mbox{ for }j=1,\cdots,N.\nonumber
\end{array}
\right. \end{equation} Here, $λ_j,\,μ_j>0$ and $β_{ij}=β_{ji}$ are constants for $i,j=1,\cdots,N$ and $i\neq j$. $B_1$ denotes the unit ball in the Euclidean space $\mathbb{R}^3$ centred at the origin. For any $P_1,\cdots,P_N\in\mathbb{N}$, we prove the uniqueness of the radial solution $(u_1,\cdots,u_j)$ with $u_j$ changes its sign exactly $P_j$ times for any $j=1,\cdots,N$ in the following case: $λ_j\geq1$ and $|β_{ij}|$ are small for $i,j=1,\cdots,N$ and $i\neq j$. New Liouville-type theorems and boundedness results are established for this purpose. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_15831 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Liouville-type theorem for the coupled Schrödinger systems and the uniqueness of the sign-changing radial solutions Li, Haoyu Miyagaki, Olímpio Hiroshi Analysis of PDEs 35A02, 35B53, 35J47 In this paper, we study the sign-changing radial solutions of the following coupled Schrödinger system \begin{equation} \left\{ \begin{array}{lr} -Δu_j+λ_j u_j=μ_j u_j^3+\sum_{i\neq j}β_{ij} u_i^2 u_j \,\,\,\,\,\,\,\, \mbox{in }B_1 ,\nonumber u_j\in H_{0,r}^1(B_1)\mbox{ for }j=1,\cdots,N.\nonumber \end{array} \right. \end{equation} Here, $λ_j,\,μ_j>0$ and $β_{ij}=β_{ji}$ are constants for $i,j=1,\cdots,N$ and $i\neq j$. $B_1$ denotes the unit ball in the Euclidean space $\mathbb{R}^3$ centred at the origin. For any $P_1,\cdots,P_N\in\mathbb{N}$, we prove the uniqueness of the radial solution $(u_1,\cdots,u_j)$ with $u_j$ changes its sign exactly $P_j$ times for any $j=1,\cdots,N$ in the following case: $λ_j\geq1$ and $|β_{ij}|$ are small for $i,j=1,\cdots,N$ and $i\neq j$. New Liouville-type theorems and boundedness results are established for this purpose. |
| title | A Liouville-type theorem for the coupled Schrödinger systems and the uniqueness of the sign-changing radial solutions |
| topic | Analysis of PDEs 35A02, 35B53, 35J47 |
| url | https://arxiv.org/abs/2401.15831 |