两类奇异椭圆问题解的存在性研究

本项目主要研究变分法和临界点理论以及非线性泛函分析中的各种技巧在探讨临界Grushin问题和临界或近乎临界Choquard问题解的存在性方面的应用，具体内容包括：1、利用有限维约化方法并结合局部Pohozaev恒等式等技巧为两类临界Grushin问题构建无穷多个新的柱状对称的多峰解；同时应用爆破分析理论和Harnack不等式等技巧探讨一类临界Grushin问题的某种多峰解的非存在性；2、借助Lusternik-Schnirelmann畴数理论和Poincare多项式等工具研究黎曼流形上的一类近乎临界和一类次临界Choquard问题的多解性；3、采用变分法、Benci伪指标理论和Nehari流形方法证明两类临界Choquard问题解的多重性。期望通过对上述问题的分析和讨论，发现并提出一些解决相关问题的新方法和新技巧，获得有意义的结果，进一步完善对此类问题的研究。

This project mainly focuses on the applications of variational methods, critical point theory and various skills in nonlinear functional analysis on the existence of solutions for critical Grushin problems and critical or almost critical Choquard problems. The specific contents include: 1. Using the finite-dimensional reduction method and the local Pohozaev identities, we construct infinitely many new cylindrically symmetric multi-bubbling solutions for two kinds of critical Grushin problems; And applying blow-up analysis technique and Harnack inequality, we discuss the nonexistence of some multi-bubbling solutions to a class of critical Grushin problems; 2. By means of Lusternik-Schnirelmann category theory and Poincare polynomial, we study the multiple solutions for a class of almost critical Choquard problems and a class of subcritical Choquard problems on Riemannian manifolds; 3. By using variational method, Benci pseudo-index theory and Nehari manifold, we prove the multiplicity of solutions for two kinds of critical Choquard problems. Hope that through the analysis and discussion on the above problems, we can find and put forward some new methods and skills to solve the related problems, obtain meaningful research results, and further improve the study on these problems.