Energy local minimizers for the nonlinear Schr\"{o}dinger equation on product spaces

TL;DR

This paper studies the existence and properties of energy minimizers with fixed $L^2$-norm for the nonlinear Schrödinger equation on product spaces, revealing conditions under which these minimizers are trivial or nontrivial along the compact manifold component.

Contribution

It extends the analysis of energy local minimizers for the nonlinear Schrödinger equation to mass-supercritical cases on product spaces, identifying when minimizers are constant or nontrivial along the compact manifold.

Findings

Existence of local minimizers for small $L^2$-mass.

Minimizers are constant along $M^k$ for sufficiently small mass.

Under certain conditions, minimizers are nontrivial along $M^k$.

Abstract

We investigate the existence of local minimizers with prescribed $L^2$-norm for the energy functional associated to the mass-supercritical nonlinear Schr\"{o}dinger 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.