两类椭圆方程解的多重性和相关性质研究

Choquard方程和Grushin算子分别是数学物理和几何分析领域的国际热点研究对象。本项目拟针对一类全空间上的Choquard方程和一类有界区域上含有Grushin算子的椭圆方程（Grushin方程）展开深入研究，主要内容包括：1、采用Benci伪指标理论证明一类带有联合非局部项的临界Choquard方程的半经典解的多重性，借助变分理论分析其基态解的收敛性、集中性和衰减性，并利用亏格理论和约束变分方法探究其正规化解的多重性，考察方程同时含有上、下临界指数时正规化基态解的存在性、非存在性及其相关性质；2、应用Lyapunov-Schmidt约化方法结合局部Pohozaev恒等式技巧，围绕一类有界区域上的超临界Grushin方程构建无穷多个新形式的高能量多峰解。上述问题的深入研究，不仅可以丰富现有的非线性椭圆方程理论体系，而且可以拓展变分理论在非线性椭圆方程中的应用范围。

Choquard equation and Grushin operator are international popular topics in the fields of mathematical physics and geometric analysis, respectively. This project intends to focus on a class of Choquard equations in the whole spaces and as well a class of elliptic equations (called Grushin equations) containing Grushin operators in bounded domains. The main contents of researches include: 1. We use Benci pseudo-index theory to prove the multiplicity of semiclassical solutions for the critical Choquard equations with combined nonlocal terms, apply the variational method to analyze the convergence, concentration and decaying properties of its ground state, utilize the genus theory and the constrained variational method to explore the multiplicity of its normalized solutions, and investigate the existence, nonexistence and properties of its normalized ground state when the equation contains the upper and lower critical exponents simultaneously; 2. Employing the Lyapunov-Schmidt reduction and the local Pohozaev identities, we construct infinitely many new types of multi-bubbled and high energy solutions for the supercritical Grushin equations in bounded domains. The deep investigations of the above problems can not only enrich the theory of nonlinear elliptic PDEs, but also extend the application ranges of the variational theory to nonlinear elliptic PDEs.