Analysis of nonlinear Schroedinger equation with electromagnetic potentials

非线性薛定谔方程不仅是一个重要的数学对象，而且是描述物理、工程等现象的重要对象。本文从数学的角度研究了一类具有磁场的非线性薛定谔方程解的多重存在性及其一些性质。更准确地说，在非线性是超线性和亚临界的情况下，当磁场具有有界井，即其底部(零点集)的非空内部时，证明了该方程有多个解，这些解位于磁井附近，并且在空间无穷远处指数衰减。

The nonlinear Schroedinger equation has not only the importance as a mathematical object but the one that describes the phenomenon in physics, engineering, etc. In this research, the multiple existence of solutions and some properties of a nonlinear Schroedinger equation with magnetic field is studied mathematically. More precisely, in the case where the nonlinearity is super-linear and subcritical and the magnetic field has the bounded well, i.e., a nonempty interior of its bottom (zero set), it is shown that the equation has multiple solutions which are localized near the magnetic well and decay exponentially at spatial infinity.