带有共振的非线性哈密顿系统多解问题的研究

Multiple solutions of nonlinear Hamiltonian systems with resonance are investigated in this project. For the sake of physical background of these problems, they were concerned by many physicians and mathematicians. We mainly study multiple solutions of multiple solutions of asymptotically linear first order and second order Hamiltonian systems with resonance. Researchers studied many problems of multiple solutions of Hamiltonian system with nonresonance. For resonant problems, former researchers got that there exists a solution mainly by degree theory. In this project, firstly, we will study the simple situation with 1 dimension, 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 firstly study the second order Hamiltonian systems. Secondly, 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.

本项目研究非线性哈密顿系统解的多重性的问题。这些问题由于有物理意义，历来受到物理学家和数学家的关注。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。而对于对于共振的情形，前人的研究大多从度的理论得到存在一个解。在这里，我们将先研究简单的一维情形，再研究高维情形。在研究高维时，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。所以我们先研究二阶哈密顿系统，再从之前的研究中借鉴一下有益的方法来研究一阶哈密顿系统。我们将使用指标理论和Morse理论，以及寻找新的方法解决上面的问题，以期得到两个或者的更多的非平凡解的存在性。

本项目研究非线性哈密顿系统解的多重性问题。我们主要研究带有共振情形非线性的一阶和二阶哈密顿系统解的多重性问题。对于1维的带有共振的二阶哈密顿系统，我们用Morse理论和指标理论得到了两个非平凡解存在的几个定理。对于高维的带有共振的非线性二阶哈密顿系统，用分歧理论和Morse理论进行研究，初步地到了一些新的结果。并在研究过程中，用指标理论对渐近线性的椭圆方程进行研究，得到了两个新的关于多解的结果且已发表。对于一阶高维哈密顿系统，由于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，解决的方法正在探索中且已取得了新的进展。我们将使用指标理论和Morse理论，以及寻找新的方法解决上面的问题，以期得到两个或者的更多的非平凡解的存在性。