带变号位势的Hamilton系统的同宿轨

In this project we will use variational methods to investigate the existence and multiplicity of homoclinic orbits for Hamiltonian systems with sign-changing potentials. Homoclinic orbits of Hamiltonian systems is very important in Mathematical physics, also is a problem of great concern in nonlinear analysis fields. In recent two decades, many domestic and foreign scholars have used nonlinear functional analysis methods to study the existence and multiplicity of homoclinic orbits for Hamiltonian systems, and obtained rich and deep results, but less results consider the Hamiltonian systems with sign-changing potentials, especially under the conditions which will be discussed in this project. In this project we will summarize intrinsic difficultiy for these problems and find new ideas in study of these problems, aim to promote the further research on these problems, and at the same time enhance people's understanding and in-depth knowledge of these important models in Mathematical physics.

本项目拟运用变分方法研究带变号位势的Hamilton系统同宿轨的存在性与多解性。Hamilton系统的同宿轨具有非常重要的数学物理意义，也是非线性分析研究领域十分关注的问题。近二十年，众多国内外学者运用非线性泛函分析方法研究其存在性与多重性，已取得了丰富而深刻的结果，但现有结果较少涉及带变号位势的情形，尤其在本项目将要讨论的几类条件下的结果几乎没有。本项目将通过分析具体问题总结这类问题的本质困难，探寻解决问题的新思路，旨在促进对这类问题的进一步关注和研究，同时加深人们对数学物理中的这类重要模型的认识和理解。

本项目分别通过改进L(t)和W(t,x)的条件，获得带变号位势Hamilton系统同宿轨新的存在性和多重性结果。具体地我们在如下三种假设下进行讨论：①去掉L(t)的周期性和强制性条件的假设②将W(t,x)超二次条件弱化为局部超二次条件③ 假设W(t,x)满足渐近二次增长条件（包括共振和非共振两种情况）。