非局部椭圆方程驻波解的存在性与动力学研究

The non-local elliptic equation has been widely used in geometry, probability theory, quantum mechanics, materials science, finance and other fields, and it is one of the most active research fields in recent 20 years. Among them, the fractional Schrödinger equation and the Choquard equation in quantum mechanics have attracted wide attention. .In this proposed research, based on the existing literature, we will use the method of variation together with the elliptic equation theory to explore some key issues for these equations. Among these issues are the existence and multiplicity of standing waves, sign changing solutions,ground state solutions and semi-classic solutions, as well as some dynamical properties of the standing waves such as centrality，decay property and regularity etc. We also expect to develop some novel and more effective techniques which will enable us to obtain some new and significant results to the theory of the non-local elliptic equations.

非局部椭圆方程在几何学、概率论、量子力学、材料学、金融等领域有着广泛的应用，是最近二十年最活跃的研究领域之一。其中，特别是量子力学中的分数阶薛定谔方程、Choquard方程受到了广泛的关注。. 本项目将使用变分法与椭圆方程理论，在已有文献的基础上，重点研究分数阶薛定谔方程与Choquard方程的核心问题：驻波解的存在性与多重性、变号解、基态解、半经典解的存在性，以及解的集中性、衰减性和正则性等动力学性态。发展非线性分析方法、技巧，深化数学工具，对所研究的问题获得若干新的、有意义的结果，推进非局部椭圆方程定性理论的发展。

非局部椭圆方程在几何学、概率论、量子力学、材料学、金融等领域有着广泛的应用，是最近二十年最活跃的研究领域之一。其中，特别是量子力学中的分数阶薛定谔方程、Choquard方程受到了广泛的关注。本项目将使用变分法与椭圆方程理论，在已有文献的基础上，重点研究分数阶薛定谔方程与Choquard方程的核心问题：驻波解的存在性与多重性、变号解、基态解、半经典解的存在性，以及解的集中性、衰减性和正则性等动力学性态。发展非线性分析方法、技巧，深化数学工具，对所研究的问题获得若干新的、有意义的结果，推进非局部椭圆方程定性理论的发展。