非局部微分方程的变分方法研究

具有变分结构的非局部微分方程有着深刻的物理背景，是非线性泛函分析研究领域的热点问题之一。项目将首先利用临界点理论等非线性分析方法研究广义Choquard方程、分数阶Choquard型椭圆方程以及Maxwell-Dirac方程组与Klein-Gordon-Dirac方程组等非局部问题驻波解的存在性并分析解的相关性质；然后研究非局部微分方程的半经典问题，通过发展非局部微分方程的扰动方法与罚函数技巧，考察其半经典解的存在性与集中性。对非局部微分方程的变分方法与技巧的系统研究，可以促进非线性泛函分析理论的发展，其研究成果可用于解释量子力学、光学等物理科学领域中的非线性现象。

Nonlocal differential equations with variaitonal structure have deep physical backgrounds and are very important in the field of nonlinear functional analysis. Firstly, the project aims to investigate the existence and analysis the properties of the standing wave solutions of the generalized Choquard equation，fractional elliptic equations related to pseudorelativistic Choquard equation and Maxwell-Dirac system and Klein-Gordon-Dirac system. Secondly, the project will also develop the pertubation methods and penalization methods to study the existence and concentration behavior of the semiclassical solutions of these nonlocal differential problems. By studing the variaitonal methods for nonlocal differential equaitons, the results of the project will help to develop the theory of nonlinear analysis and explain the nonlinear phenomenas in quantum mechanics and optics in physics.