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We study the focusing mass-subcritical biharmonic nonlinear Schr\"odingerequation (BNLS) on the product space $\mathbb{R}_x^d\times\mathbb{T}_y^n$.Following the crucial scaling arguments introduced in \cite{TTVproduct2014} weestablish existence and stability results for the normalized ground states ofBNLS. Moreover, in the case where lower order dispersion is absent, we provethe existence of a critical mass number $c_0\in(0,\infty)$ that sharplydetermines the $y$-dependence of the deduced ground states. In the mixeddispersion case, we encounter a major challenge as the BNLS is no longerscale-invariant and the arguments from \cite{TTVproduct2014} for determiningthe sharp $y$-dependence of the ground states fail. The main novelty of thepresent paper is to address this difficult and interesting issue: Using adifferent scaling argument, we show that $y$-independence of ground states withsmall 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 massmust possess non-trivial $y$-dependence by appealing to some novel constructionof test functions. The latter particularly holds for all parameters lying inthe full mass-subcritical regime.

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