一类稳态Schödinger-Poisson-Slater方程标准化解的研究

项目名称： 一类稳态Schödinger-Poisson-Slater方程标准化解的研究

项目编号： No.11501137

项目类型： 青年科学基金项目

立项/批准年度： 2016

项目学科： 数理科学和化学

项目作者： 罗庭健

作者单位： 广州大学

项目金额： 18万元

中文摘要： 稳态的Schödinger-Poisson-Slater方程源于量子多体系统的研究，是一类含非局部项的稳态非线性Schödinger方程。本项目主要研究该方程一类特殊解的存在性及其相关性质。该类解的L2-范数是给定的，我们称之为标准化解（Normalized solutions），这类解具有一定的物理意义。方程的标准化解对应于其能量泛函在约束流形下的临界点。此时方程中的参数λ则以Lagrange乘数的形式出现。目前国内外对该类方程标准化解的研究结果并不多，本项目拟在已经获得的部分研究结果基础上，采用约束变分理论，结合山路定理等方法，深入研究该类方程在不同非线性条件下其标准化解的存在性及其性质，特别是解的存在性对相关参数的依赖关系。

中文关键词： 变分方法；Schödinger-Poisson-Slater方程；标准化解；存在性

英文摘要： Stationary Schödinger-Poisson-Slater equations stem from the study of systems of many particles in Quantum Mechanics, which are a class of nonlinear Schrödinger equations with nonlocal term. This program is devoted to study the existence and related properties of one special kind of solutions to the equations. Such kinds of solutions whose L2 norm is prescribed, are called normalized solutions, which have some physical background. Normalized solutions to the equations correspond to critical points of their energy functional on a constrained manifold. In such case, the parameter λ in the equations is not fixed any more and appears as a Lagrange Multiplier. Up to the moment, the studies of normalized solutions for Schödinger-Poisson-Slater equations are rare. Based on the existing results, we shall applying the constrained variational methods, Mountain Pass Theorem and related methods, study further the existence of normalized solutions of Schödinger-Poisson -Slater equations with different nonlinearities. In particular, the relationship between the existence and related parameters will be investigated.

英文关键词： variational method;Schödinger-Poisson-Slater equations;normalized solutions;existence