几类临界Choquard方程的变分方法研究

The nonlinear Choquard equation arises in various fields of mathematical physics. In the last decades, a great deal of mathematical efforts has been devoted to the study of existence, multiplicity and properties of the solutions of the nonlinear Choquard equation. In this project, we study the existence and multiplicity of solution for the critical Choquard equations by variational methods, the critical point theory and analytical techniques. Firstly, we study the existence of Mountain-Pass solution via a concentration-compactness principle for the Choquard equation. Secondly, we also study the existence of high energy solution by using a global compactness lemma for the nonlocal Choquard equation. By applying minimax procedure and perturbation technique, we is to obtain the existence of infinitely many solutions. Lastly, by assuming that the potential might be sign-changing, we study the existence and multiplicity of semiclassical states for the critical Choquard equation by the Mountain-Pass Lemma and the genus theory.

非线性Choquard方程具有广泛的物理背景，在近几十年里得到了学者们的广泛关注，致力于解的存在性，多解性以及解的性质方面的研究。项目将利用变分方法，临界点理论及分析技巧研究几类临界Choquard方程解的存在性和多解性。首先，我们通过非局部情形下的集中紧性原理拟证明临界Choquard方程山路解的存在性。其次，通过全局集中紧性原理考虑Choquard方程高能量解的多解性。我们还利用极小极大方法和扰动技巧拟证明方程无穷多解的存在性。最后，考虑临界Choquard方程的半经典问题，方程的位势函数可以是变号的，并通过山路引理和亏格理论拟证明方程高能量半经典解的存在性和多解性。

首先，研究了几类临界Choquard方程的半经典问题，位势函数可以是变号的或者是非负的、亦或是不变号的。在适当的条件下，我们通过山路定理和亏格理论证明了方程半经典解的存在性和多解性。其次，研究了一类耦合临界Choquard方程组问题，我们建立了非局部情形下的全局集中紧性原理，并证明了高能量解的存在性。最后，利用Lyapunov-Schmidt 约化方法和Pohožaev 恒等式研究了具有对称或非对称位势函数的非线性Choquard方程的无穷多解性。