一类Schrodinger-Poisson型方程解的存在性与集中行为

Schrodinger-Poisson equation is an very important field equation,which is widely used in quantum mechanics,semiconductor theory and statistical mechanics.This project focuses on a class of Schrodinger-Poisson equation with attractive nonlocal term and satisfying Berestycki-Lions condition,which is a generalization of the Choquard equation,it can be used to model nonlocal problem.We consider the existence and concentration behavior of solutions.Our research is divided into two aspects:(1)For sub critical case,we consider the existence and concentration behavior of solutions for Schrodinger-Poisson equation under Berestycki-Lions condition,including the existence of solutions and the existence of multipeak solutions;(2)For critical case,including Sobolev critical case and Hardy-Littlewood-Sobolev critical case,we consider the existence and concentration behavior of solutions for Schrodinger-Poisson equation under Berestycki-Lions condition.To solve the above problems,we need to overcome the difficulties caused by nonlocal items and Berestycki-Lions conditions.Through the research of this project,we can deeply understand the influence of non local term and Berestycki-Lions conditions on the properties of the solutions.

Schrodinger-Poisson方程是一类重要的场方程,在量子力学、半导体理论、统计力学中有广泛应用.本项目围绕一类非线性项满足Berestycki-Lions条件且非局部项为吸引项的Schrodinger-Poisson型方程展开,该方程是Choquard方程的推广,可用于描述非局部问题.本项目关注解的存在性与集中行为.研究分为两方面:(1)非线性项满足次临界Berestycki-Lions条件时解的存在性与集中行为,包括解的多重性与多峰解;(2)非线性项满足临界Berestycki-Lions条件(包括Sobolev临界和Hardy-Littlewood-Sobolev临界)时解的存在性与集中行为.如何克服非局部项和Berestycki-Lions条件产生的困难是解决问题的关键。通过研究可以深入认识非局部项和Berestycki-Lions条件对解的相关性质产生的影响。

Schrodinger-Poisson型方程具有很强物理背景，是与量子动力学密切相关的一类数学模型。本项目主要围绕两类Schrodinger-Poisson型方程展开，利用变分理论、罚函数方法等手段研究了解的存在性与多解问题，并对相关不等式进行了深入分析。详细研究工作如下：1.利用罚函数方法，对非线性项满足Berestycki-Lions条件的二维Choquard方程解的半经典极限问题进行了分析，得到其集中行为。2. 利用变分理论研究了一类带有非局部卷积项以及混合位势的P-Choquard型方程解的存在性。3.利用辅助函数以及精细的能量估计，得到Gross-Pitaevskii方程基态解的爆破行为。4.利用散度定理，对球面上Hardy不等式以及Heisenberg 群上Leray-Trudinger不等式进行了深入分析。