On existence and stability results for normalized ground states of mass-subcritical biharmonic NLS on $\mathbb{R}^d\times\mathbb{T}^n$

TL;DR

This paper investigates the existence and stability of normalized ground states for the mass-subcritical biharmonic nonlinear Schrödinger equation on product spaces, addressing challenges in scale-invariance and $y$-dependence with novel scaling and construction techniques.

Contribution

It introduces new scaling arguments and test function constructions to analyze ground state properties, especially in the mixed dispersion case where previous methods fail.

Findings

Existence and stability of ground states established for certain regimes.

Identification of a critical mass $c_0$ influencing $y$-dependence.

Proof that small mass ground states are $y$-independent, large mass states are not.

Abstract

We study the focusing mass-subcritical biharmonic nonlinear Schr\"odinger equation (BNLS) on the product space $\mathbb{R}_x^d\times\mathbb{T}_y^n$. Following the crucial scaling arguments introduced in \cite{TTVproduct2014} we establish existence and stability results for the normalized ground states of BNLS. Moreover, in the case where lower order dispersion is absent, we prove the existence of a critical mass number $c_0\in(0,\infty)$ that sharply determines the $y$-dependence of the deduced ground states. In the mixed dispersion case, we encounter a major challenge as the BNLS is no longer scale-invariant and the arguments from \cite{TTVproduct2014} for determining the sharp $y$-dependence of the ground states fail. The main novelty of the present paper is to address this difficult and interesting issue: Using a different scaling argument, we show that $y$-independence of ground states with small mass still holds in the case $\beta>0$ and $\alpha\in(0,4/(d+n))$. Additionally, we also prove that ground states with sufficiently large mass must possess non-trivial $y$-dependence by appealing to some novel construction of test functions. The latter particularly holds for all parameters lying in the full mass-subcritical regime.