Energy local minimizers for the nonlinear Schrödinger equation on product spaces
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909663139725312 |
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| author | Pierotti, Dario Verzini, Gianmaria Yu, Junwei |
| author_facet | Pierotti, Dario Verzini, Gianmaria Yu, Junwei |
| contents | We investigate the existence of local minimizers with prescribed $L^2$-norm for the energy functional associated to the mass-supercritical nonlinear Schrödinger equation on the product space $\mathbb{R}^N \times M^k$, where $(M^k,g)$ is a compact Riemannian manifold, thus complementing the study of the mass-subcritical case performed by Terracini, Tzvetkov and Visciglia in [\emph{Anal. PDE} 2014, arXiv:1205.0342].
First we prove that, for small $L^2$-mass, the problem admits local minimizers. Next, we show that when the $L^2$-norm is sufficiently small, the local minimizers are constants along $M^k$, and they coincide with those of the corresponding problem on $\mathbb{R}^N$. Finally, under certain conditions, we show that the local minimizers obtained above are nontrivial along $M^k$. The latter situation occurs, for instance, for every $M^k$ of dimension $k\ge 2$, with the choice of an appropriate metric $\hat g$, and in $\mathbb{R}\times\mathbb{S}^k$, $k\ge 3$, where $\mathbb{S}^k$ is endowed with the standard round metric. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_22371 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Energy local minimizers for the nonlinear Schrödinger equation on product spaces Pierotti, Dario Verzini, Gianmaria Yu, Junwei Analysis of PDEs We investigate the existence of local minimizers with prescribed $L^2$-norm for the energy functional associated to the mass-supercritical nonlinear Schrödinger equation on the product space $\mathbb{R}^N \times M^k$, where $(M^k,g)$ is a compact Riemannian manifold, thus complementing the study of the mass-subcritical case performed by Terracini, Tzvetkov and Visciglia in [\emph{Anal. PDE} 2014, arXiv:1205.0342]. First we prove that, for small $L^2$-mass, the problem admits local minimizers. Next, we show that when the $L^2$-norm is sufficiently small, the local minimizers are constants along $M^k$, and they coincide with those of the corresponding problem on $\mathbb{R}^N$. Finally, under certain conditions, we show that the local minimizers obtained above are nontrivial along $M^k$. The latter situation occurs, for instance, for every $M^k$ of dimension $k\ge 2$, with the choice of an appropriate metric $\hat g$, and in $\mathbb{R}\times\mathbb{S}^k$, $k\ge 3$, where $\mathbb{S}^k$ is endowed with the standard round metric. |
| title | Energy local minimizers for the nonlinear Schrödinger equation on product spaces |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2506.22371 |