论文标题
在$ \ mathbb {r}^d \ times \ mathbb {t}^n $上，在质量批判性biharmonic nls的归一化基态的存在和稳定性结果上
On existence and stability results for normalized ground states of mass-subcritical biharmonic NLS on $\mathbb{R}^d\times\mathbb{T}^n$
论文作者
论文摘要
我们研究了产品空间上的焦点质量质量biharmonic biharmonic非线性schrödinger方程（bnls）$ \ mathbb {r} _x^d \ times \ times \ times \ mathbb {t} _y^n $。在\ cite {ttvproduct2014}中引入的关键缩放参数之后，我们为BNL的归一化基态建立了存在和稳定性结果。此外，在不存在较低阶段的情况下，我们证明存在（0，\ infty）$的临界质量数$ C_0 \，这些\ in（0，\ infty）$急剧确定了推论基态的$ y $依赖性。在混合分散案例中，我们遇到了一个重大挑战，因为BNL不再是规模不变，并且来自\ cite {ttvproduct2014}的论点是确定地面状态的尖锐$ y $依赖性失败。本文的主要新颖性是要解决这个困难而有趣的问题：使用不同的规模论点，我们表明，$ y $的基态在$β> 0 $和$α\ in（0,4/（d+n））$中仍然存在。此外，我们还证明，质量足够大的基态必须通过吸引一些新的测试功能结构来具有非平凡的$ y $依赖性。后者尤其适合所有参数，位于整个质量临界状态下。
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 $β>0$ and $α\in(0,4/(d+n))$. 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.
