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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2502.08912 |
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| _version_ | 1866909490720276480 |
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| author | Li, Haoyu Miyagaki, Olímpio Hiroshi Wang, Zhi-Qiang |
| author_facet | Li, Haoyu Miyagaki, Olímpio Hiroshi Wang, Zhi-Qiang |
| contents | We investigate the global bifurcation structure of the radial nodal solutions to the coupled elliptic equations \begin{equation}
\left\{
\begin{array}{lr}
-Δu+u=u^3+βuv^2\mbox{ in }B_1 ,\nonumber
-Δv+v=v^3+βu^2v\mbox{ in }B_1 ,\nonumber
u,v\in H_{0,r}^1(B_1).\nonumber
\end{array}
\right. \end{equation} Here $B_1$ is a unit ball in $\mathbb{R}^3$ and $β\in\mathbb{R}$ the coupling constant is used as bifurcation parameter. For each $k$, the unique pair of nodal solutions $\pm w_k$ with exactly $k-1$ zeroes to the scalar field equation $-Δw + w=w^3$ generate exactly four synchronized solution curves and exactly four semi-trivial solution curves to the above system. We obtain a fairly complete global bifurcation structure of all bifurcating branches emanating from these eight solution curves of the system, and show that for different $k$ these bifurcation structures are disjoint. We obtain exact and distinct nodal information for each of the bifurcating branches, thus providing a fairly complete characterization of nodal solutions of the system in terms of the coupling. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_08912 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global bifurcations of nodal solutions for coupled elliptic equations Li, Haoyu Miyagaki, Olímpio Hiroshi Wang, Zhi-Qiang Analysis of PDEs 35B32, 35J47 We investigate the global bifurcation structure of the radial nodal solutions to the coupled elliptic equations \begin{equation} \left\{ \begin{array}{lr} -Δu+u=u^3+βuv^2\mbox{ in }B_1 ,\nonumber -Δv+v=v^3+βu^2v\mbox{ in }B_1 ,\nonumber u,v\in H_{0,r}^1(B_1).\nonumber \end{array} \right. \end{equation} Here $B_1$ is a unit ball in $\mathbb{R}^3$ and $β\in\mathbb{R}$ the coupling constant is used as bifurcation parameter. For each $k$, the unique pair of nodal solutions $\pm w_k$ with exactly $k-1$ zeroes to the scalar field equation $-Δw + w=w^3$ generate exactly four synchronized solution curves and exactly four semi-trivial solution curves to the above system. We obtain a fairly complete global bifurcation structure of all bifurcating branches emanating from these eight solution curves of the system, and show that for different $k$ these bifurcation structures are disjoint. We obtain exact and distinct nodal information for each of the bifurcating branches, thus providing a fairly complete characterization of nodal solutions of the system in terms of the coupling. |
| title | Global bifurcations of nodal solutions for coupled elliptic equations |
| topic | Analysis of PDEs 35B32, 35J47 |
| url | https://arxiv.org/abs/2502.08912 |