An extension of Cabré-Chanillo theorem to the $p$-laplacian
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866917056577798144 |
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| author | Grossi, Massimo Montoro, Luigi Sciunzi, Berardino Wang, Zexi |
| author_facet | Grossi, Massimo Montoro, Luigi Sciunzi, Berardino Wang, Zexi |
| contents | In this paper, we study the critical points of stable solutions for the following $p$-laplacian equation \begin{equation*} \begin{cases} -div\big(|\nabla u|^{p-2}\nabla u\big)=f(u)&in\ \Om,\\ u>0&in\ \Om,\\ u=0&on\ \partial\Om, \end{cases} \end{equation*} where $p>2$, $f\in C^1([0,+\infty))$ satisfies $f(t)>0$ for $t>0$, and $\Om\subset\R^2$ is a smooth bounded domain with non-negative curvature of the boundary. Via a suitable approximation argument, we prove that, a stable solution $u$ admits, as its only critical point, the internal absolute maxima and possibly saddle points with zero index. Moreover, $Argmax(u)$ is a point or segment. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_01542 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An extension of Cabré-Chanillo theorem to the $p$-laplacian Grossi, Massimo Montoro, Luigi Sciunzi, Berardino Wang, Zexi Analysis of PDEs 35J92, 35B38, 35B09 In this paper, we study the critical points of stable solutions for the following $p$-laplacian equation \begin{equation*} \begin{cases} -div\big(|\nabla u|^{p-2}\nabla u\big)=f(u)&in\ \Om,\\ u>0&in\ \Om,\\ u=0&on\ \partial\Om, \end{cases} \end{equation*} where $p>2$, $f\in C^1([0,+\infty))$ satisfies $f(t)>0$ for $t>0$, and $\Om\subset\R^2$ is a smooth bounded domain with non-negative curvature of the boundary. Via a suitable approximation argument, we prove that, a stable solution $u$ admits, as its only critical point, the internal absolute maxima and possibly saddle points with zero index. Moreover, $Argmax(u)$ is a point or segment. |
| title | An extension of Cabré-Chanillo theorem to the $p$-laplacian |
| topic | Analysis of PDEs 35J92, 35B38, 35B09 |
| url | https://arxiv.org/abs/2511.01542 |