对称群作用下非线性Choquard方程变号解的存在性及其性质

The current project is devoted to the studies on the existences and properties of nodal solutions for the nonlinear Choquard equation defined on the N-dimensional Euclidean space under the symmetric group actions. Nonlinear Choquard equations have become one of hot topics in the field of nonlinear functional analysis in the past few decades due to its profound backgrounds and wide applications in mathematics and physics. By means of symmetric group actions, the sign-changing solutions constructed in this project demonstrate the nonlocal feature of this equation. Such type of nodal solutions admit hyperplanes as their nodal sets and have complicated symmetric structure and exact nodal domains, while the locally nonlinear Schrödinger equations do not possess this kind of sign-changing solutions. Nodal solutions will be constructed by establishing the local compactness conditions for the energy functionals in the non-compact variational framework under the symmetric group actions. The main results of this project can be stated as follows. 1) the Riesz type Choquard equation admits sign-changing solutions with high energies; 2) the nonlinear logarithmic Choquard equation has nodal solutions with complicated symmetric structures; 3) the nonlinear Choquard equation possesses sign-changing solutions with prescribed symmetric structures and L^2 norm. These results will promote the frontier research of nodal solutions for this nonlocal equation and enrich the existence theory for weak solutions, and further expand the scope of application of variational methods and the critical point theory in solving nonlocal nonlinear problems.

本项目研究对称群作用下N维欧氏空间上非线性Choquard方程变号解的存在性及其相关性质。非线性Choquard方程因其深厚的数学物理应用，已成为非线性分析领域的热点研究课题。借助于（有限）对称群作用，本项目所构造的变号解刻画了此类方程的非局部特性：其零点集全是超平面，且具有一定的对称结构和明确的变号区域数。与其联系紧密的局部非线性Schrödinger方程却不可能具有这一类型的变号解。本项目拟引入对称群作用，在非紧的变分框架下建立能量泛函的局部紧性条件来构造变号解。证明的主要结果有：1）Riesz型Choquard方程具有高能量变号解；2）对数型Choquard方程存在高对称性的变号解；3）L^2模约束的非线性Choquard方程存在具有指定对称性的变号解。这些结果将促进此类方程变号解的前沿研究，发展和完善弱解的存在性理论，进一步拓展变分法和临界点理论在解决非局部非线性问题中的应用范围。

作为一类典型的非局部非线性问题，非线性Choquard方程在数学与物理中均有着广泛的应用。正是由于它的非局部特性，此类方程有着与局部方程本质不同的非线性现象，已成为近年来非线性分析领域的热点研究课题。.本项目主要研究N维欧氏空间上非线性Choquard方程变号解的存在性及其相关性质。主要利用非线性泛函分析理论与方法特别是变分方法、临界点理论、对称临界原理以及集中紧性原理来研究Choquard方程变号解的存在性以及解的对称性和变号节点数等性质。本项目取得的研究进展如下：1）证明了非线性Choquard方程马鞍型变号解的存在性，包括次临界Choquard方程、上临界Choquard方程、具一般非线性项的Choquard方程以及分数阶Choquard方程等情形；2）证明了非线性Choquard方程高拓扑半经典解的多重存在性与渐近性态；3）建立了非线性Choquard方程马鞍型规范解的存在性，包括质量超临界和质量次临界情形。.通过建立新的分析技巧与方法，本项目建立了紧性缺失情形下运用Coxeter对称群构造Choquard方程马鞍型变号解、马鞍型规范解的变分理论。项目的研究结果丰富了Choquard方程变号解的存在性理论，进一步刻画了Choquard方程的非局部特性。本项目的研究有利于推动此类方程变号解的前沿进展，发展和完善非局部方程弱解的存在性理论，推进非线性分析理论与非线性方程应用的发展。