两类非局部薛定谔方程变号解的存在性及其渐近性态研究

The study of sign-changing solutions of nonlocal Schrödinger equations is very complicated and challenging due to the presence of the nonlocal term which leads to plenty of difficulties that the corresponding energy functional dose not satisfy “additivity”and that energy of sign-changing solutions is not easy to be estimated and so on. The existing studies mainly focus on cases concerning some special nonlinear terms. Based on comprehensive applications of modern mathematics knowledge, we aim to develop theory of sign-changing solutions of nonlocal Schrödinger equations, with emphasis on the existence, concentrating behavior, exponential decay of sign-changing solutions for two classes of nonlocal Schrödinger equations(Schrödinger-Poisson equations and Kirchhoff-type equations) with a general type of nonlinear terms that maybe not satisfy Ambrosetti-Rabnowitz condition or monotonicity condition, in order to reveal the influence of nonlocal term on the topological properties of solution, and to promote the development of existence theory of nonlocal Schrödinger equations, and to build a solid theoretical foundation and provide some efficient methods in solving more complicated nonlocal problems. The study of sign-changing solutions of nonlocal Schrödinger equations not only needs the classic theory of differential equations, but also calls for other related knowledge including topology, functional analysis and so on, cannot only enrich the qualitative theory of nonlocal differential equations, but also contribute to novel ideas, new theory and technical methodology in mathematics such as the nonlinear analysis. Most importantly, the study of sign-changing solutions of nonlocal differential equations involves the interaction of different branches of mathematics disciplines, and should be of a great importance in both mathematical theory and applications.

非局部薛定谔方程变号解问题是一类非常有挑战性的研究课题。非局部项的出现使得能量泛函不具有“可加性”，变号解能量难以估计等，因此，变号解问题研究非常复杂。现有研究工作主要针对一些比较特殊的非线性项展开。本项目拟综合运用现代数学知识，发展非局部薛定谔方程变号解理论，重点研究两类具有一般非线性项（特别是A-R 条件或者单调性条件不成立的情形）的非局部薛定谔方程（薛定谔泊松方程和基尔霍夫方程）变号解的存在性、集中性、指数衰减性等等，揭示非局部项对解的拓扑性质所产生的影响，推动非局部薛定谔方程解的存在性理论的发展，为研究更加复杂的非局部问题提供一套比较系统的理论和研究方法；该项目研究既要用到经典的微分方程理论，又要用到拓扑、泛函分析等相关知识，不仅可丰富非局部微分方程定性理论知识，又可探索数学（尤其是非线性分析）及其交叉应用中的新思想、新理论和新方法，且可使不同数学分支学科之间进行相互交叉与渗透。

本项目自立项以来，课题组对非局部薛定谔方程变号解的存在性及其渐近性态问题进行了研究，获得了一系列显著的研究成果，且都发表在国际知名SCI源刊杂志上。这些成果为整个项目的顺利完成打下了良好的基础，也达到了本项目所制定的预期目标。本项目的研究成果主要包括两个方面，具体可归结如下： .在理论研究方面，发展了一种非局部扰动方法，并结合极大极小方法以及动力系统中的流不变集概念研究了非局部薛定谔方程的变号解问题，得到了一系列存在性与多重性的研究成果；发展了非局部极值原理，研究了非局部薛定谔方程变号基态解的存在性、部分对称性及其渐近性质；另外，发展新的局部扰动方法研究了非局部薛定谔方程无穷多非平凡解的存在性。.在应用研究方面，利用以上研究的Dancer-Fucik谱性质研究了具有非共振非线性项的非局部薛定谔方程变号解的存在性问题。此外，发展了新的能量估计方法去研究时滞微分方程整体解的存在性，并用来证明多智能体系统的渐近集群与编队。