On the existence of degenerate solutions of the two-dimensional $H$-system
We consider entire solutions $\omega \in \dot{H}^1(\mathbb{R}^2;\mathbb{R}^3)$ of the $H$-system \begin{equation*} \Delta \omega =2\omega _x\wedge \omega _y\, \end{equation*} which we refer to as bubbles. Surprisingly, and contrary to conjectures raised in the literature, we find that bubbles with...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Académie des sciences
2025-04-01
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| Series: | Comptes Rendus. Mathématique |
| Subjects: | |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.731/ |
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| Summary: | We consider entire solutions $\omega \in \dot{H}^1(\mathbb{R}^2;\mathbb{R}^3)$ of the $H$-system
\begin{equation*}
\Delta \omega =2\omega _x\wedge \omega _y\,
\end{equation*}
which we refer to as bubbles. Surprisingly, and contrary to conjectures raised in the literature, we find that bubbles with degree at least three can be degenerate: the linearized $H$-system around a bubble can admit solutions that are not tangent to the smooth family of bubbles. We then give a complete algebraic characterization of degenerate bubbles. |
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| ISSN: | 1778-3569 |