强不定问题的变分方法  

Variational methods for strongly indefinite problems

作　　者：丁彦恒[1,2] 

机构地区：[1]中国科学院数学与系统科学研究院,北京100190 [2]中国科学院大学数学科学学院,北京100190

出　　处：《中国科学：数学》2017年第7期779-810,共32页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:10831005;11331010;10421001和10640420049)资助项目

摘　　要：本文概述作者承担的国家自然科学基金项目所获得的部分成果,特别是从源头出发系统地培育强不定问题变分方法的特色方向,并开启一些应用问题的研究,包括(1)建立强不定问题的变分框架的基本方法;(2)建立局部凸拓扑线性空间的形变理论,相应得到处理强不定问题的临界点定理;(3)首次研究非自治稳态Dirac系统解的存在性,特别是突破强不定困难获得其半经典解的存在性、集中现象和指数衰减性;(4)首次得到非线性(非自治、无界Hamilton型)反应-扩散系统整体解的存在性和多重性,特别是奇异扰动下其基态解的存在性、集中现象和衰减性;(5)深入研究Hamilton系统的同宿轨和Schrdinger方程的全局解;(6)其他初始性工作,如自旋流形上的Dirac方程的分歧现象.This paper reviews some recent development in studying variational methods for strongly indefinite problems under the support of National Natural Science Foundation of China. We firstly describe the ideas for establishing the variational setting, then state a deformation theory in locally convex topological linear spaces and several critical point theorems for dealing with strongly indefinite problems based on the deformation. Some applications of the theory are given including mainly： Existence and multiplicity of solutions to non-autonomous stationary Dirac system, in particular, the existence, concentration phenomena and the exponential decay of semi-classical states; existence and multiplicity of global solutions to the nonlinear（non-autonomous, unbounded Hamilton-type） reaction-diffusion system, especially the existence, concentration phenomena and decay of its ground states of the system with singular perturbation; depth study of homoclinic orbits Hamiltonian system,global Schrdinger equation; other initial work, such as bifurcation of Dirac equation on spin manifolds.

关 键 词：强不定问题 变分框架 形变理论 DIRAC 系统 无界 HAMILTON 自旋流形 

分 类 号：O175[理学—数学] O177[理学—基础数学]