非周期Hamilton系统同宿轨的研究

Hamiltonian systems is an important field of research in nonlinear science. It comes from the geometrical optics, and has a natural connection with classical mechanics and celestial mechanics. In this project, the variational methods and critical point theory are applied to study the existence and multiplicity of homoclinic solutions of nonperiodic Hamiltonian systems. The main contents are the following: .(1) We will study the existence and multiplicity of homoclinic solutions of the first-order nonperiodic Hamiltonian systems with superquadratic、asymptotically quadratic or locally subquadratic potentials..(2) We will investigate the existence of infinitely many homoclinic solutions of the second-order nonperiodic systems with concave-convex nonlinearity or without symmetry. Futhermore, we will study the existence and multiplicity of homoclinic solutions of the second-order nonperiodic systems with sign-changing potentials. The effect of different nonlinear conditions, jointly with the change of parameter, on the existence of homoclinic solutions and on the number of slutions will be discussed..(3) We will study the existence and multiplicity of homoclinic solutions of the singular nonperiodic second-order Hamiltonian systems, and investigate the existence of multibump homoclinic solutions..We will analyse and summarize the common features of the above problems and new difficulties, and explore new research methods. We hope to make some contributions to the development of nonlinear analysis via the study of this project.

Hamilton系统是非线性科学的一个重要研究领域。它源于几何光学，同经典力学和天体力学有着自然联系。本项目拟使用变分方法和临界点理论研究非周期Hamilton系统同宿轨的存在性和多重性，主要内容包括：（1）研究一阶非周期Hamilton系统分别在超二次、渐近二次和局部次二次情形下同宿轨的存在性、多重性以及无穷多条同宿轨的存在性。（2）研究二阶非周期Hamilton系统，当位势函数为一般凹凸非线性项或者不具有对称性时无穷多条同宿轨的存在性；研究具有变号位势的二阶非周期系统同宿轨的存在性和多重性，揭示不同的非线性条件以及参数变化对同宿轨的存在性和存在个数的影响。（3）研究高维情形下二阶奇异非周期Hamilton系统同宿轨的存在性和多重性，讨论在其周围是否存在多重碰撞型同宿轨。我们将分析和总结这类问题的共同特点和新的困难，探索新的研究方法。期望本课题的研究能促进非线性分析理论及其应用的发展。

本项目使用变分法和临界点理论研究了Hamilton系统同宿轨的存在性和多重性，以及无界区域上非局部椭圆问题基态解的存在性。研究了一类的二阶周期Hamilton系统当位势函数在无穷远处超二次但不必满足Ambrosetti-Rabinowitz条件时基态同宿轨的存在性；研究了一类特殊的二阶Hamilton系统，当位势函数满足不同增长性条件时，同宿轨的存在性和多重性；研究了一类非周期带阻尼项微分方程在次二次或者渐近二次增长情形下快同宿轨的存在性、多重性以及无穷多条快同宿轨的存在性。此外，研究了一类带临界指数的非自治Schrodinger-Poisson系统非平凡解的存在性、集中性和基态解的存在性；带临界指数的非自治Kirchhoff-type方程基态解的存在性。丰富和推广了已有的结果。在项目执行期间，主要参加了非线性分析国际会议暨第二十届全国非线性泛函分析会议和非线性分析与反应扩散方程国际研讨会。