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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2506.22371 |
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Table of 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.