On the critical points of solutions of Robin boundary problems
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2024
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| _version_ | 1866910597938937856 |
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| author | De Regibus, Fabio Grossi, Massimo |
| author_facet | De Regibus, Fabio Grossi, Massimo |
| contents | In this paper we prove the uniqueness of the critical point for stable solutions of the Robin problem \[ \begin{cases} -Δu=f(u)&\text{in }Ω\\ u>0&\text{in }Ω\\ \partial_νu+βu=0&\text{on }\partialΩ, \end{cases} \] where $Ω\subseteq\mathbb{R}^2$ is a smooth and bounded domain with strictly positive curvature of the boundary, $f\ge0$ is a smooth function and $β>0$. Moreover, for $β$ large the result fails as soon as the domain is no more convex, even if it is very close to be: indeed, in this case it is possible to find solutions with an arbitrary large number of critical points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_06576 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the critical points of solutions of Robin boundary problems De Regibus, Fabio Grossi, Massimo Analysis of PDEs In this paper we prove the uniqueness of the critical point for stable solutions of the Robin problem \[ \begin{cases} -Δu=f(u)&\text{in }Ω\\ u>0&\text{in }Ω\\ \partial_νu+βu=0&\text{on }\partialΩ, \end{cases} \] where $Ω\subseteq\mathbb{R}^2$ is a smooth and bounded domain with strictly positive curvature of the boundary, $f\ge0$ is a smooth function and $β>0$. Moreover, for $β$ large the result fails as soon as the domain is no more convex, even if it is very close to be: indeed, in this case it is possible to find solutions with an arbitrary large number of critical points. |
| title | On the critical points of solutions of Robin boundary problems |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2409.06576 |