奇异摄动非线性薛定谔方程组的相关研究

In this project, some properties of solutions of singular perturbed nonlinear Schrodinger systems will be studied by Lyapunov-Schmidt reduction method and variation techniques base on some well-known works. These contents are listed as following:.(I)For a class of singular perturbed nonlinear Schrodinger systems defined on bounded domains with Neumann boundary conditions, the existence, multiplication and the estimation on the number of spike solutions will be considered in this project. These spike solutions are those solutions with interior spikes, solutions with boundary spikes and solutions with interior and boundary spikes..(II) For a class of singular perturbed nonlinear Schrodinger systems defined on whole space with some potentials, when the potentials have positive lower bounded, the existence of infinitely many segregated solutions will be considered in this project; when the potentials are critical frequency, i.e., the potentials are nonnegative with nonempty zero sets, the existence of solutions with multi-scale bumps will also be considered in this project.

本项目在已有文献的基础上采用Lyapunov-Schmidt约化方法和变分技巧进一步深入研究奇异摄动非线性薛定谔方程组解的相关性质，主要包括：.(I)对于有界区域上Neumann边界条件下奇异摄动非线性薛定谔方程组，本项目主要研究多峰解的存在性，多重性以及多峰解的数量估计. 峰解主要包括内峰解、边界峰解以及含有内峰与边界峰的混合峰解..(II)对于全空间上带位势的奇异摄动非线性薛定谔方程组，当位势函数具有正下界时，本项目主要研究无穷多个分离解的存在性；当位势函数是临界位势（非负且零点集不空）时，本项目主要研究多尺度峰解的存在性.

本项目在已有文献的基础上采用Lyapunov-Schmidt约化方法和变分技巧进一步深入研究奇异摄动非线性薛定谔方程组解的相关性质，主要包括： (I)对于有界区域上Neumann边界条件下奇异摄动非线性薛定谔方程组，本项目主要研究多 峰解的存在性，多重性以及多峰解的数量估计. 峰解主要包括内峰解、边界峰解以及含有内 峰与边界峰的混合峰解. (II)对于全空间上带位势的奇异摄动非线性薛定谔方程组，当位势函数具有正下界时，本项目主要研究无穷多个分离解的存在性；当位势函数是临界位势（非负且零点集不空）时，本项目主要研究多尺度峰解的存在性...除此之外，本项目还研究了一些其它相关问题，如带线性耦合项的奇异摄动非线性薛定谔方程组解的存在性与多重性、Cosserat 能量泛函临界点的边界正则性、.和Grushin-型方程柱状解的存在性与多重性等。