非线性Schordinger方程及其相关问题的变分方法研究

项目名称： 非线性Schordinger方程及其相关问题的变分方法研究

项目编号： No.11471235

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 黄毅生

作者单位： 苏州大学

项目金额： 69万元

中文摘要： 本项目拟应用变分方法和临界点理论研究一类非线性Schrodinger方程及其相关的椭圆方程的解的存在性和解的性态问题。这类方程起源于量子物理，它在非线性光学、电磁学、凝聚态物理等领域中有着许多重要的应用，是当今非线性分析领域的研究热点，吸引了众多数学研究者的兴趣，涌现出了大量突出的研究成果，同时也留下了一些极具挑战性的课题。本项目的研究包含了下面三个方面的内容： (1)建立研究算子和非线性项都是强不定时Schrodinger方程解的存在性和多重性的一般方法，探索不同的位势函数对方程解的存在性与解的性态的影响； (2)研究在位势变号及位势在无穷远处消失时，Schrodinger-Poisson系统的可解性和多解性及解的性态； (3)研究有界或无界域上一类奇异椭圆方程及其耦合方程组的可解性、多解性及其解的性态，并结合增广拉格朗日函数方法等优化技巧寻求一些约束优化问题的可解性。

中文关键词： Schrodinger方程；边值问题；变分方法；临界点理论；多解

英文摘要： In this project we will study the existence of solutions and properties of the solutions for a class of nonlinear Schrodinger equations and its relevant elliptic equations via variational methods and critical point theory. Arised from quantum physics，this class of Schrodinger equations have many important applications in nonlinear optics, electromangetics, condensed matter physics, etc, and become a hot spot of current research in the field of nonlinear functional analysis. The equations have attracted the interest of many mathematical researchers, and obtained a large number of outstanding research achievements, as well as many challenging research topics are left. This project contains the following three aspects: (1) We try to set up the general method for the study of the existence and multiplicity of Schrodinger equations with strongly indefinite potentials and nonlinear terms, and explore the influences of various potentials on the existence of solutions and properties of the solutions for the equations; (2) We will study the solvability, the multiplicity and the properties of solutions for a class Schrodinger-Poisson system with sign-changing potentials or potentials vanishing at infinity; (3) We will discuss the solvability, the multiplicity and the properties of solutions for a class of singular elliptic equations and couple systems on bounded or unbounded domains, and we will also study the solvability of some constrained optimization problems by combining with the optimization techniques such as the augmented Lagrangian function method, etc.

英文关键词： Schordinger equation;bouandary value problem;variational method;critical point theory;multiple solutions