现代变分理论的若干问题研究

Variational methods are important research areas in mathematics worldwide. The objects of variational methods are nonlinear differential equations with variational structure. These equations arise from mathematical physics, bio-engineering, economic theory and other disciplines, and therefore are of theoretical importance and a wide background in applications. This project is to gather researchers who work in variational methods so that they can collaborate in research both in abstract theories and in applications. The plan of this project is to apply various variational and topological methods, including minimax methods，Morse theory, index theory, bifurcation theory and minimization methods, to study several important topics in nonlinear analysis. More precisely, we will use locally convex topologies to construct deformation between different level sets, and to develop critical point theory for strongly indefinite functionals, we will develop critical point theory for infinite dimensional Hamiltonian systems, quasilinear elliptic equations and nonlinear Schrodinger systems, and we will study existence of solutions, multiplicity of solutions, classification of solutions, and analytical, geometrical, and topological properties of solutions for nonlinear Dirac equations, quasilinear elliptic equations, semi-linear Schrodinger equations and systems, and Hamiltonian systems which have variational structures. Through the study of these topics, we will manage to add new ingredients, new ideas and new methods to nonlinear variational theories, and to make important breakthroughs in research in theoretical and applied nonlinear analysis.

变分法是国内外数学研究的重要领域。变分法的研究对象是具有变分结构的非线性微分方程，这些方程来自数学物理、生物工程、经济理论等学科，具有重要的理论意义和广泛的应用背景。本项目集中团队的整体力量，在理论和应用两方面开展合作研究，拟应用极大极小方法、Morse理论、指标理论、分歧理论、极小化方法等变分和拓扑方法，研究非线性分析中若干前沿课题。具体地说，拟利用局部凸拓扑建立泛函的水平集之间的形变，并发展强不定泛函的临界点理论；拟针对无穷维哈密顿系统、拟线性椭圆型方程和非线性薛定谔方程组发展相应的临界点理论；拟研究非线性Dirac方程、拟线性椭圆型方程、半线性薛定谔方程（组）、Hamilton系统等具有变分结构的微分方程解的存在性、多重性、解的类型、解的分析性质、几何性质和拓扑性质等。通过研究这些问题，本项目将为变分理论注入新内容、新思想和新方法，力争在非线性分析理论与应用的研究中有重要突破。

我们发挥团队的整体力量，按计划执行， 获得理想成果。已正式发表论文83余篇。论文被引用率显著提升，据美国数学会MathSciNet网站显示，2013/07/05—2018/12/9项目组成员（7人）论文被引用4226次(2013/07/05统计数2832次、2018/12/9统计数7058次)。项目组成员组织3个定期学术讨论班，培养42名研究生（23名博士、15名硕士、4名博士后）。. 变分法是国际上数学研究的重要领域，研究对象是具有变分结构的非线性微分方程。这些方程来自数学物理、生物工程、经济理论等科学领域，具有重要的理论和应用背景。本项目研究的重要成果主要包含下面6个方面：（1）一个重大突破 (Dirac方程半经典解的存在性与集中现象)；（2）一个特色方向成形（强不定问题的变分方法）；（3）不可压欧拉方程定常涡解的研究；（4）多方面推广和改进了Clark定理、椭圆型偏微分方程变号解的研究；（5）波方程和Monge-Ampere方程组；（6）多项非线性问题及其他问题的重要进展.