Threshold solutions for the $3d$ cubic INLS: the energy-critical case
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866917191365951488 |
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| author | Campos, Luccas Farah, Luiz Gustavo Murphy, Jason |
| author_facet | Campos, Luccas Farah, Luiz Gustavo Murphy, Jason |
| contents | We study the energy-critical $3d$ cubic inhomogeneous NLS equation $i\partial_t u + Δu + |x|^{-1}|u|^2 u=0$. In this work, we prove the existence of special solutions $W^\pm$ with energy equal to that of the ground state $W$ and use these solutions to characterize the behavior of solutions at the ground state energy. The singular factor $|x|^{-1}$ in the nonlinearity significantly limits the smoothness of the ground state and prompts a novel approach to the modulation analysis. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_05349 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Threshold solutions for the $3d$ cubic INLS: the energy-critical case Campos, Luccas Farah, Luiz Gustavo Murphy, Jason Analysis of PDEs We study the energy-critical $3d$ cubic inhomogeneous NLS equation $i\partial_t u + Δu + |x|^{-1}|u|^2 u=0$. In this work, we prove the existence of special solutions $W^\pm$ with energy equal to that of the ground state $W$ and use these solutions to characterize the behavior of solutions at the ground state energy. The singular factor $|x|^{-1}$ in the nonlinearity significantly limits the smoothness of the ground state and prompts a novel approach to the modulation analysis. |
| title | Threshold solutions for the $3d$ cubic INLS: the energy-critical case |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2601.05349 |