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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/2502.05876 |
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| _version_ | 1866916853419343872 |
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| author | Manabe, Kanako Tanaka, Satoshi |
| author_facet | Manabe, Kanako Tanaka, Satoshi |
| contents | \begin{equation*}
\left\{
\begin{array}{l}
u'' + λh(x,α) e^u = 0, \quad x \in (-1,1), \\[1ex]
u(-1) = u(1) = 0,
\end{array}
\right. \end{equation*} where $λ>0$, $0<α<1$, $h(x,α)=0$ for $|x|<α$, and $h(x,α)=1$ for $α\le |x| \le 1$. We compute the Morse index of positive even solutions, and then we prove the existence of an unbounded connected set of positive non-even solutions emanating from a symmetry-breaking bifurcation point. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_05876 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Morse index and symmetry-breaking bifurcation of positive solutions to the one-dimensional Liouville type equation with a step function weight Manabe, Kanako Tanaka, Satoshi Analysis of PDEs 34B18, 34C23, 35B32 \begin{equation*} \left\{ \begin{array}{l} u'' + λh(x,α) e^u = 0, \quad x \in (-1,1), \\[1ex] u(-1) = u(1) = 0, \end{array} \right. \end{equation*} where $λ>0$, $0<α<1$, $h(x,α)=0$ for $|x|<α$, and $h(x,α)=1$ for $α\le |x| \le 1$. We compute the Morse index of positive even solutions, and then we prove the existence of an unbounded connected set of positive non-even solutions emanating from a symmetry-breaking bifurcation point. |
| title | Morse index and symmetry-breaking bifurcation of positive solutions to the one-dimensional Liouville type equation with a step function weight |
| topic | Analysis of PDEs 34B18, 34C23, 35B32 |
| url | https://arxiv.org/abs/2502.05876 |