Kirchhoff型拟线性Schrodinger方程及其耦合系统的非光滑变分方法研究

In this program, we shall study the existence and multiplicity of standing waves for quasi-linear Schrodinger equations of Kirchhoff type and non semi-trivial solutions for their couling system via variational methods. This is a class of nonlocal problems. It was proposed as a model for free vibrations of elastic strings and can model the population density. Moreover, its degenerative local form appeared in the study of plasma physis and condensed matter theory. First, we shall study the variational structures for the problems. Generally, because of the reason of quasi-linearity and non locality, there are variational functionals for the problems, formally. But, our working spaces are not vector spaces, only completed metric spaces. Hence classical critical point theory and minimax method can not be applied, and hence we shall study the program from the following three ways:. (1) The existence and multiplicity of ground state solutions will be studied by Nehari method and constrained minimization arguments；. (2) The study of the problems will be changed into corresponding semi-linear problems by a dual approach and a perturbation approach;. (3) The existence and multiplicity of non trivial(or non semi-trivial) weak solutions and multibump solutions will be studied by non smooth critical point theory for continuous functional in completed metric spaces.

本项目拟采用变分方法研究Kirchhoff型的拟线性Schrodinger方程的驻波解及其耦合系统的非半平凡解的存在性和多重性问题。 这是一类非局部问题，起源于一个有弹性细绳的自由振动模型，还可模拟动物种群的稠密性，并且，其退化的局部形式亦出现于等离子物理和凝聚态理论的研究中。本项目首先将研究其变分结构。一般来说，由于拟线性和非局部性的原因，此类问题仅只是在形式上有变分泛函，其工作空间甚至不是线性空间，仅仅是完备度量空间，传统的临界点理论和极大极小方法不适用于其研究。因此，本项目将从下面三方面进行研究：.（1）采用Nehari方法和受约束的极小化讨论研究其基态解的存在性和多重性；.（2）采用对偶方法和扰动方法将原问题转化为相应的半线性问题进行研究；.（3）应用近年发展起来的完备度量空间上连续泛函的非光滑临界点理论研究其非平凡（或非半平凡）弱解和多胞解的存在性和多重性。

本项目采用变分方法研究Kirchhoff型的拟线性Schrodinger方程的驻波解及其耦合系统的非平凡解的存在性和多重性问题。该问题是一个非局部问题，起源于一个有弹性细绳的自由振动模型，还可模拟动物种群的稠密性，并且，其退化的局部形式亦出现于等离子物理和凝聚态理论的研究中。本项目主要采用Nehari流形方法、对偶方法、扰动方法和完备度量空间中连续泛函的非光滑临界点理论研究其非平凡弱解、基态解、多胞解、几何相异解、高能解、低能解、小解和变号解的存在性及多重性问题。此外，本项目还作了一些相关问题的扩展研究。