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

Title: 关于质量次临界双调和NLS在$\mathbb{R}^d\times\mathbb{T}^n$上归一化基态的存在性和稳定性结果

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. Show Chinese abstract

Abstract: 我们研究在乘积空间$\mathbb{R}_x^d\times\mathbb{T}_y^n$上的聚焦质量次临界双调和非线性薛定谔方程（BNLS）。 遵循在\cite{TTVproduct2014}中引入的关键尺度论证，我们建立了 BNLS 的归一化基态的存在性和稳定性结果。 此外，在低阶色散缺失的情况下，我们证明了一个临界质量数$c_0\in(0,\infty)$的存在，该数严格决定了推导出的基态的$y$依赖性。 在混合色散情况下，我们遇到了一个主要挑战，因为 BNLS 不再具有尺度不变性，而来自\cite{TTVproduct2014}的用于确定基态的尖锐$y$依赖性的论证失效。 本文的主要创新之处在于解决这个困难而有趣的问题：通过不同的缩放论证，我们证明了在情况$\beta>0$和$\alpha\in(0,4/(d+n))$下，小质量的基态仍保持$y$独立性。 此外，我们还证明了通过引入一些新颖的测试函数构造，足够大质量的基态必须具有非平凡的$y$依赖性。 后者尤其适用于全部质量亚临界区域内的所有参数。