分数阶椭圆方程与哈密顿系统多解问题的研究

项目名称： 分数阶椭圆方程与哈密顿系统多解问题的研究

项目编号： No.11471067

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： 常小军

作者单位： 东北师范大学

项目金额： 74万元

中文摘要： 本项目致力于应用变分与拓扑方法研究分数阶椭圆方程和哈密顿系统的多解问题。随着在反常扩散、非牛顿流体力学、量子力学 、粘弹性力学、软物质物理力学、美式期权等领域研究的需要，对分数阶微分方程的研究越来越得到人们的广泛关注，而对分数阶椭圆方程的研究正成为当前非线性分析领域的一个研究热点。哈密顿系统作为一类重要的数学模型一直受到人们的高度重视，其中奇异哈密顿系统由于其特殊而重要的背景，更是受到众多著名数学家的关注和深入研究。 本项目主要研究：1）有界区域上分数阶拉普拉斯方程边值问题解的多重性；2）分数阶薛定谔方程的基态解的存在性以及束缚态解的多重性,进一步研究其变号解的存在性与多重性；3）奇异哈密顿系统的固定能量问题解的多重性；4）带有阻尼项的奇异哈密顿系统解的存在性与多重性。

中文关键词： 变分法；拓扑度理论；分数阶椭圆方程；哈密顿系统；多重性

英文摘要： This project is devoted to the study of multiplicity of solutions of fractional elliptic equations and Hamiltonian systems by using variational and topological methods. Along with the development of several areas such as anomalous diffusion, non-Newtonian fluid mechanics, quantum mechanics, viscoelastic mechanics, soft matter physics, American options in finance, the people are paying more and more attention to fractional differential equations, and the study of fractional elliptic equations has been being a popular subject in the nonliear analysis area. On the other hand, as an important model, Hamiltonian systems has alaways been highly valued by many famous mathematicians, while the issue of singular Hamiltonian systems has gotten widespread attention and has been gotten many research achievements due to its special background. The research plan of this project includes the following subject areas: 1)Multiplicity of solutions of boundary value problems for fractional Laplacian equations in bounded domains; 2)Existence of ground states and multiplicity of bound states of fractional Schrodinger equations, specially the existence and multiplicity of sign-changing solutions; 3)Multiplicity of solutions of fixed energy problems of singular Hamiltonian systems; 4)Existence and multiplicity of sulutions of singular Hamilonian systems with damping term.

英文关键词： Variational method;Topological degree theory;Fractional elliptic equations;Hamiltonian systems;Multiplicity