强不定问题的变分方法
发布时间：2018-12-05 21:01
【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銉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.
[Abstract]:This paper summarizes some of the achievements of the National Natural Science Foundation project undertaken by the author, especially the characteristic direction of systematically cultivating the variational methods for strongly indefinite problems from the source, and opens the research on some application problems. The main contents are as follows: (1) the basic method of establishing a variational framework for strongly indefinite problems; (2) the deformation theory of locally convex topological linear space is established, and the critical point theorem for dealing with strongly indefinite problems is obtained. (3) the existence of solutions for nonautonomous steady-state Dirac systems is studied for the first time, especially the existence, concentration and exponential decay of semi-classical solutions are obtained by breaking through the strong indeterminacy. (4) for the first time, we obtain the existence and multiplicity of global solutions for nonlinear (nonautonomous, unbounded Hamilton type) reaction-diffusion systems, especially the existence, concentration and attenuation of ground state solutions under singular perturbations; (5) the global solutions of homoclinic orbits and Schr's dinger equations for Hamilton systems are studied in depth, and (6) other initiality work, such as bifurcation of Dirac equations on spheroidal manifolds.
【作者单位】： 中国科学院数学与系统科学研究院;中国科学院大学数学科学学院;
【基金】：国家自然科学基金(批准号:10831005,11331010,10421001和10640420049)资助项目
【分类号】：O175;O177
本文编号：2365465
[Abstract]:This paper summarizes some of the achievements of the National Natural Science Foundation project undertaken by the author, especially the characteristic direction of systematically cultivating the variational methods for strongly indefinite problems from the source, and opens the research on some application problems. The main contents are as follows: (1) the basic method of establishing a variational framework for strongly indefinite problems; (2) the deformation theory of locally convex topological linear space is established, and the critical point theorem for dealing with strongly indefinite problems is obtained. (3) the existence of solutions for nonautonomous steady-state Dirac systems is studied for the first time, especially the existence, concentration and exponential decay of semi-classical solutions are obtained by breaking through the strong indeterminacy. (4) for the first time, we obtain the existence and multiplicity of global solutions for nonlinear (nonautonomous, unbounded Hamilton type) reaction-diffusion systems, especially the existence, concentration and attenuation of ground state solutions under singular perturbations; (5) the global solutions of homoclinic orbits and Schr's dinger equations for Hamilton systems are studied in depth, and (6) other initiality work, such as bifurcation of Dirac equations on spheroidal manifolds.
【作者单位】： 中国科学院数学与系统科学研究院;中国科学院大学数学科学学院;
【基金】：国家自然科学基金(批准号:10831005,11331010,10421001和10640420049)资助项目
【分类号】：O175;O177
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