分数阶Klein-Gordon问题和相关变分法的研究

This proposal intends to study the problems about Klein-Gordon fields in quantum field theory. We mainly consider Klein-Gordon-Maxwell systems, including fractional cases and critical cases. The fractional cases in our proposal is the fractional Laplacian acting on the spatial variables...We intend to find new nonlinear analysis methods and techniques related to partial differential equations to solve the Klein-Gordon problems including fractional cases. The purpose of the proposal is establishing the existence, uniqueness, multiplicity and correlation properties of solutions to the Klein-Gordon problems, including solitary wave solutions, standing wave solutions and ground state solutions. Among the tools involved, we mention (nonlocal) concentration compactness principle, minimax priciple, index theory and methods related to partial differential equation etc.

本项目拟研究量子场论中有关Klein-Gordon场的问题，主要考虑Klein-Gordon-Maxwell方程组，包括分数阶情形和临界情形。本项目中的分数阶指的是作用在空间变量上的分数阶Laplacian。..我们拟寻找新的非线性分析方法和偏微分方程有关技巧，使之适用于解决分数阶情形的Klein-Gordon问题，建立Klein-Gordon相关问题解的存在性、惟一性、多重性和相关性态，包括孤立波解、驻波解、基态解等。非局部集中紧方法、极大极小原理、指标理论以及偏微分方程相关方法等将会被涉及到。