Strong instability of standing waves for $L^2$-supercritical Schrödinger-Poisson system with a doping profile
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913860289560576 |
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| author | Colin, Mathieu Watanabe, Tatsuya |
| author_facet | Colin, Mathieu Watanabe, Tatsuya |
| contents | This paper is devoted to the study of the nonlinear Schrödinger-Poisson system with a doping profile. We are interested in the strong instability of standing waves associated with ground state solutions in the $L^2$-supercritical case. The presence of a doping profile causes several difficulties, especially in examining geometric shapes of fibering maps along an $L^2$-invariant scaling curve. Furthermore, the classical approach by Berestycki-Cazenave for the strong instability cannot be applied to our problem due to a remainder term caused by the doping profile. To overcome these difficulties, we establish a new energy inequality associated with the $L^2$-invariant scaling and adopt the strong instability result developed by Fukaya-Ohta(2018). When the doping profile is a characteristic function supported on a bounded smooth domain, some geometric quantities related to the domain, such as the mean curvature, are responsible for the strong instability of standing waves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_03991 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Strong instability of standing waves for $L^2$-supercritical Schrödinger-Poisson system with a doping profile Colin, Mathieu Watanabe, Tatsuya Analysis of PDEs 35J20, 35B35, 35B44, 35Q55 This paper is devoted to the study of the nonlinear Schrödinger-Poisson system with a doping profile. We are interested in the strong instability of standing waves associated with ground state solutions in the $L^2$-supercritical case. The presence of a doping profile causes several difficulties, especially in examining geometric shapes of fibering maps along an $L^2$-invariant scaling curve. Furthermore, the classical approach by Berestycki-Cazenave for the strong instability cannot be applied to our problem due to a remainder term caused by the doping profile. To overcome these difficulties, we establish a new energy inequality associated with the $L^2$-invariant scaling and adopt the strong instability result developed by Fukaya-Ohta(2018). When the doping profile is a characteristic function supported on a bounded smooth domain, some geometric quantities related to the domain, such as the mean curvature, are responsible for the strong instability of standing waves. |
| title | Strong instability of standing waves for $L^2$-supercritical Schrödinger-Poisson system with a doping profile |
| topic | Analysis of PDEs 35J20, 35B35, 35B44, 35Q55 |
| url | https://arxiv.org/abs/2411.03991 |