带有共振的渐近线性哈密顿系统周期解的存在性和多重性问题的研究

项目名称： 带有共振的渐近线性哈密顿系统周期解的存在性和多重性问题的研究

项目编号： No.11501178

项目类型： 青年科学基金项目

立项/批准年度： 2016

项目学科： 数理科学和化学

项目作者： 李科强

作者单位： 河南师范大学

项目金额： 18万元

中文摘要： 本项目研究带有共振的渐近线性哈密顿系统周期解的存在性和多重性问题。哈密顿系统是非线性科学研究中的重要领域，由于物理以及工程中的很多模型都是以哈密顿系统或其扰动形式出现的。多年来，该领域的研究是重要的课题。我们主要研究带有共振情形渐近线性的一阶和二阶哈密顿系统解的存在性和多重性问题。之前人们研究了很多带有非共振的哈密顿系统的多解问题。而对于共振的情形，前人的研究大多从度的理论进行研究。后来, 为了得到更好地结果，人们引入很强的条件来研究共振的问题。在这里, 我们将在更一般的条件下, 先研究简单的一维达芬方程情形, 再研究高维情形。研究高维时，由于一阶对应的泛函为强不定泛函，其对应的Morse指标为无穷，研究起来难度较大。所以我们再先研究二阶, 再从之前的研究中借鉴有益的方法来研究一阶。我们将用指标理论和Morse理论，以及寻找新方法解决问题，以期得到两个或更多非平凡解。

中文关键词： 哈密顿系统；指标理论；；渐近线性；；共振；解的多重性

英文摘要： Existence and multiplicity of periodic solutions for asymptotically linear Hamiltonian systems with resonance are investigated in this program. Since many models in physics and engineering are all presented in the form of the Hamiltonian system or disturbance, Hamiltonian system is an important field in nonlinear science research. So for many years, the studies in this field are the important programs. We mainly study existence and multiplicity of solutions for asymptotically linear first order and second order Hamiltonian systems with resonance. Researchers studied many problems of multiple solutions of Hamiltonian system with nonresonance ago. For resonant problems, former researchers mainly use degree theory to study them. Later, people studied the resonant problems under the strong conditions to obtain the better results. In this program, under more general conditions, firstly, we will study the simple situation with 1 dimension i.e. Duffing equations, and then investigate the problems with high dimensions. Since the functional of first order system with high dimensions is strong indefinite and its Morse index is infinite, it is difficult for us to solve the problems. So we successively study the second order Hamiltonian systems. At last, drawing on the experience of the useful methods in former research, we investigate the first order Hamiltonian systems. Using index theory, Morse theory, and searching for new methods to deal with problems mentioned above, we expect to obtain existence of two or more than two nontrivial solutions.

英文关键词： Hamiltonian systems;index theory ;asymptotically linear ;resonance;existence and multiplicity of solutions