Solitary waves for the power degenerate NLS -- existence and stability
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866914980579770368 |
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| author | Iyer, Vishnu Stefanov, Atanas G. |
| author_facet | Iyer, Vishnu Stefanov, Atanas G. |
| contents | We consider a semilinear Schrödinger equation, driven by the power degenerate second order differential operator $\nabla\cdot (|x|^{2a} \nabla), a\in (0,1)$. We construct the solitary waves, in the sharp range of parameters, as minimizers of the Caffarelli-Kohn-Nirenberg's inequality. Depending on the parameter $a$ and the nonlinearity, we establish a number of properties, such as positivity, smoothness (away from the origin) and almost exponential decay. Then, and as a consequence of our variational constrcution, we completely characterize the spectral stability of the said solitons. We pose some natural conjectures, which are still open -- such as the radiality of the ground states, the non-degeneracy and most importantly uniqueness. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_15138 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Solitary waves for the power degenerate NLS -- existence and stability Iyer, Vishnu Stefanov, Atanas G. Analysis of PDEs We consider a semilinear Schrödinger equation, driven by the power degenerate second order differential operator $\nabla\cdot (|x|^{2a} \nabla), a\in (0,1)$. We construct the solitary waves, in the sharp range of parameters, as minimizers of the Caffarelli-Kohn-Nirenberg's inequality. Depending on the parameter $a$ and the nonlinearity, we establish a number of properties, such as positivity, smoothness (away from the origin) and almost exponential decay. Then, and as a consequence of our variational constrcution, we completely characterize the spectral stability of the said solitons. We pose some natural conjectures, which are still open -- such as the radiality of the ground states, the non-degeneracy and most importantly uniqueness. |
| title | Solitary waves for the power degenerate NLS -- existence and stability |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2410.15138 |