标题： NLS在乘积空间上的基态 显示英文标题

标题： The NLS ground states on product spaces

摘要： 我们研究非线性薛定谔方程在乘积空间$\R^n\times M^k$上的基态性质，其中$M^k$是一个紧致黎曼流形。 我们证明对于小的$L^2$质量，基态与相应的$\R^n$基态一致。 我们还证明在临界质量以上，基态具有非平凡的$M^k$依赖性。 最后，我们讨论了柯西问题，这将变分分析转化为动力稳定性结果。 显示英文摘要

摘要： We study the nature of the Nonlinear Schr\"odinger equation ground states on the product spaces $\R^n\times M^k$, where $M^k$ is a compact Riemannian manifold. We prove that for small $L^2$ masses the ground states coincide with the corresponding $\R^n$ ground states. We also prove that above a critical mass the ground states have nontrivial $M^k$ dependence. Finally, we address the Cauchy problem issue which transform the variational analysis to dynamical stability results.