Minimax理论、分支理论及多解问题

On this projection we plan to study some new problems of nonlinear analysis, including:.(1) We intend to find new Minimax methods, extend our Nehari type linking method built up to more general cases, and apply them to the existence of multiple solution problem associated with the Schrödinger equation, Hamiltonian system and Dirac equation. Combining the Minimax method, Morse theory, the perturbation method presented by Poincare-Hopf theorem, implicit function theorem along with the related bifurcation methods, we try to find new ways related to the multiple solution problem. To make new progress on the existence of infinitely many solutions of superlinear elliptic problem without the symmetric hypothesis, which has been an open problem for several decades..(2) Discuss about the spectrum of nonlinear Schrödinger operator and the existence of multiple solution problem associated with the Schrödinger equation with more general potential, involving the case with magnetic field.

本项目拟研究非线性分析中一些新问题，具体内容包括：.（1）寻求新的Minimax方法，把我们最近建立的Nehari型环绕方法进一步拓展并将其应用于Schrödinger方程，Hamilton系统，Dirac方程的多解存在性问题。结合Minimax方法，Morse理论，Poincare-Hopf定理中的扰动方法，隐函数定理及相应的分支方法，寻求解决多解问题的新途径。希望对破解非对称超线性椭圆型方程是否存在无穷多解，这一具有几十年历史的公开问题取得新突破。.（2）讨论非线性Schrödinger算子的谱，并对更一般位势情形讨论Schrödinger方程多解存在性，包括带磁场的Schrödinger方程解的存在性和多解问题。

本项目拟研究非线性分析中一些新问题，具体内容包括：.（1）寻求新的minimax方法，希望对破解非对称超线性椭圆型方程是否存在无穷多解，这一具有几十年历史的公开问题取得新突破。.（2）把我们建立的Nehari型环绕方法进一步拓展并将其应用于Schrödinger方程，Hamilton系统，Dirac方程的多解存在性问题。通过利用Poincaré-Hopf定理中的扰动方法拓展Morse理论，并结合隐函数定理及相应的分支方法，寻求解决多解问题的新途径。.（3）讨论非线性Schrödinger算子的谱，并针对更一般位势情形研究Schrödinger方程多解存在性，包括带磁场的Schrödinger方程解的存在性和多解问题。