论文标题
带有恒定磁场的3D非线性Schrödinger方程
The 3D nonlinear Schrödinger equation with a constant magnetic field revisited
论文作者
论文摘要
在本文中，我们对具有恒定磁场的三维非线性schrödinger方程进行了凯奇问题。我们首先建立足够的条件，以确保全球存在和有限的时间爆破解决方案。特别是，我们为全球存在而得出了尖锐的阈值，而不是质量至关重要的非线性方程式。接下来，我们证明了归一化驻波的存在和轨道稳定性，这将先前的已知结果扩展到了质量关键和质量危机的病例。为了显示出归一化孤立波的存在，我们提出了一种新的方法，避免了著名的浓度 - 紧凑原则。最后，我们研究了基态立波的存在和强烈的不稳定，从而大大改善了先前的文献。
In this paper, we revisit the Cauchy problem for the three dimensional nonlinear Schrödinger equation with a constant magnetic field. We first establish sufficient conditions that ensure the existence of global in time and finite time blow-up solutions. In particular, we derive sharp thresholds for global existence versus blow-up for the equation with mass-critical and mass-supercritical nonlinearities. We next prove the existence and orbital stability of normalized standing waves which extend the previous known results to the mass-critical and mass-supercritical cases. To show the existence of normalized solitary waves, we present a new approach that avoids the celebrated concentration-compactness principle. Finally, we study the existence and strong instability of ground state standing waves which greatly improve the previous literature.
