项目名称: Hamilton系统和几类重要椭圆方程的研究
项目编号: No.11471267
项目类型: 面上项目
立项/批准年度: 2015
项目学科: 数理科学和化学
项目作者: 唐春雷
作者单位: 西南大学
项目金额: 76万元
中文摘要: 我们拟利用临界点理论结合拓扑度方法和各种分析工具研究以下几类重要的非线性变分问题:1. 一阶、二阶Hamilton系统同宿轨的存在性、多重性以及无穷多条同宿轨的存在性;2.具有Hardy项和Hardy-Sobolev临界指数的奇异椭圆方程正解的存在性和多重性;3.Kirchhoff-type非局部问题多解、变号解的存在性以及解的几何、分析和拓扑性态;4、Euler方程的稳态解。本项目的选题具有深刻的物理、几何和生物学背景,有重要的理论意义和研究价值。我们期望通过对上述具体问题的研究,推动非线性分析理论和应用的发展,同时为其它相关科学领域提供理论参考和技术支持。
中文关键词: Hamilton系统;同宿轨;非线性椭圆方程;正解;变分与拓扑方法
英文摘要: We try to apply critical point theory, together with topological methods and various technologies in analysis, to study the following important nonlinear variational problems: 1. Existence of one nontrivial homoclinic orbit, and also infinitely many homoclinic orbits for the first (or second) order Hamiltonian systems; 2. Existence and multiplicity of positive solutions of singular elliptic equations involving Hardy terms and critical Hardy-Sobolev exponents; 3. Existence of multiple solutions and sign-changing solutions of the nonlocal Kirchhoff-type problems, and properties of solutions; 4. Stationary solutions of Euler equations. These topics have deep background of physics, geometry and biology, and therefore are important both in theory and in applications. Via the study of this project, we hope to make some contributions to the development of nonlinear analysis, and provide theoretic references and technical supports for other scientific researches.
英文关键词: Hamiltonian system;homoclinic orbits;nonlinear elliptic equations;positive solutions;variational and topological methods