具有非局域或强不定特性的非线性问题的变分和非变分方法

具有非局域或强不定特性的非线性微分方程是非线性分析领域的重要研究课题，它们有的与数学领域的重要基本问题相关，有的与物理科学中的微分方程模型有联系。项目计划研究由(加权)Hardy-Littlewood-Sobolev不等式与Sobolev嵌入的结合而衍生出来的非局域Lane-Emden方程(组)、与Hartree-Fork模型有关的非局域Choquard方程(组)、与相对论量子力学理论有关的强不定Dirac方程。项目计划在变分或非变分框架下使用非线性分析方法研究非局域或强不定特性对上述问题解的基本性质、不同类型解的分类、不同类型解的存在性与多值性的影响。项目的研究成果将有助于推动具有非局域或强不定特性的非线性偏微分微分方程的理论发展，帮助人们更好地理解数学和物理中的非线性分析问题。

Nonlinear differential equations with non-local nonlinearities or strongly indefinite structures are important research topics in nonlinear analysis. They are closely related not only to other important problems in mathematical fields, but also to the differential equation models in physics. The topics in this proposal are: non-local Lane-Emden equation(system) arising from the weighted Hardy-Littlewood-Sobolev inequalities and Sobolev imbedding; non-local Choquard equation(system) related to Hartree-Fork model; the strongly indefinite Dirac equation from relativistic quantum mechanics theory. By applying nonlinear analysis techniques under variational or non-variational structure, we are able to study the properties and classification of the solutions, the existence and multiplicity results for different types of solutions. The results obtained in this proposal will help people to better understand the nonlinear analysis problems in both fields of mathematics and physics.