临界点理论中的几个问题

Critical point theory is one of the main theories in nonlinear functional analysis and an important research area in modern mathematics as well, with very extensive and deep applications in areas such as nonlinear differential equations. In this project we will use critical point theory combined with topological methods, bifurcation theory and various technologies in analysis to study several important problems in nonlinear elliptic equations. The contents of study include mainly: existence and multiplicity of sign-changing solutions, especially existence of sign-changing solutions with prescribed number of nodal domains, and geometrical and topological properties of sign-changing solutions for boundary value problems of nonlinear elliptic partial differential equations; existence of multi-bump type sign-changing solutions and existence of multi-peak type sign-changing solutions concentrating at a single point for nonlinear Schrodinger equations; properties of solutions, especially existence and uniqueness of ground state, multiplicity of bound states, and geometrical properties of solutions for a class of nonlinear Schrodinger systems stemming from condensed matter physics and nonlinear optics and other physical areas. Those problems listed above are not only highly important mathematical problems, being located at the forefront of the research area of nonlinear analysis at the international level, but also problems with great difficulties, new methods and ideas being needed in order to solve them. We hope that, by implementing this project, several important problems in nonlinear differential equations be solved and new methods of research in the field of critical point theory be developed.

临界点理论是非线性泛函分析中主要理论之一，是现代数学的重要研究领域，在非线性微分方程等领域有非常广泛而深刻的应用。本项目将应用临界点理论并结合拓扑方法、分歧理论以及各种分析工具，对非线性椭圆型方程中几类重要问题进行研究。研究内容主要包括：非线性椭圆型偏微分方程边值问题的变号解的存在性和个数、特别是具有指定变号域个数的变号解的存在性、以及变号解的几何和拓扑性质；非线性薛定谔方程的多包型变号解的存在性和集中于一点的多峰变号解的存在性；来自凝聚态物理学和非线性光学等领域的一类薛定谔方程组的解的性质，特别是基态的存在性和唯一性、束缚态的多重性和解的几何性质等。这些问题既是重要的数学问题，处于国际非线性分析领域的研究前沿，同时又是非常困难的问题，解决起来需要新的方法和思路。通过这一研究计划，一方面期望解决非线性微分方程中的几个重要问题，另一方面期望在临界点理论中发现新的研究方法。

本项目应用临界点理论研究非线性微分方程。主要研究内容有：临界点理论、非线性椭圆型方程和方程组无穷多个解的存在性、非线性哈密尔顿系统无穷多个周期解的存在性。本项目获得了下列研究成果：从应用的角度出发，把临界点理论中经典的Clark定理做了一系列推广和改进，证明了多个新的临界点定理；证明从物理等学科中推演出来的一类椭圆型方程组有唯一正解；通过构造一类新的伪梯度和下降流不变集，证明一类非线性Schrodinger-Poisson方程组有无穷多个变号解。在证明这些结论的过程中，我们提出了多个新方法，这些方法已经被学术同行使用。