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

Abstract

We study the focusing mass-subcritical biharmonic nonlinear Schrödinger 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.

The authors welcome any comments