Energy local minimizers for the nonlinear Schrödinger equation on product spaces

Abstract

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.

22 pages