项目名称： 一类典型非线性薛定谔方程组及相关问题的研究

项目编号： No.11501143

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： 彭艳芳

作者单位： 贵州师范大学

项目金额： 18万元

中文摘要： 非线性薛定谔方程是量子力学中的基本方程，描述了非线性光学，水波和 Bose-Einstein凝聚态等许多物理现象。由于薛定谔方程具有广泛的物理应用背景，因而近年来一直是偏微分方程研究中关注的热点问题。. 本项目将重点研究一类典型的带临界指数的非线性薛定谔方程组及其相关问题，应用Nehari流形，集中紧性原理，构造函数等方法，拟将重点探讨以下几方面的问题：.(1) 低维情况下方程组极小能量正解的存在性问题；.(2) 低维情况下方程组变号解的存在性问题；.(3) 全空间中方程组极小能量解的存在性及唯一性，极小能量解流形的非退化性等问题；.(4) 方程组在非平凡拓扑区域上正解的存在性等问题。同时在上述研究的基础上，进一步探讨分数阶薛定谔方程组的一些相关问题。本项目的研究具有一定的物理意义，同时是对现有一些关于非线性薛定谔方程的重点研究成果的进一步深化探讨。

中文关键词： 薛定谔方程；临界指数；变分方法；解的存在性；非退化性

英文摘要： Nonlinear Schrödinger equation is a basic equation in quantum mechanics and is widely used in nonlinear optics, wave and Bose-Einstein condensed-matter physics model etc. In recent years, due to its wide range of physical application, Schrödinger equation has been a hot topic in the study of partial differential equations. .Our project will mainly focus on a class of typical nonlinear Schrödinger system with Sobolev critical exponents and its related problems, by virtue of Nehari manifold, concentrated compactness principle and the method of constructing functions,some problems will be studied from the following aspects: . (1) the existence of least energy positive solutions in lower dimension; . (2) the existence of sign-changing solutions in lower dimension;. (3) the existence of positive solutions, the existence and uniqueness of the least energy solution, the nondegeneracy of the mainflod of least energy solutions in the whole space; . (4) the existence of positive solutions in the nontrivial topology domains. Moreover, basing on this basis, some related problems on fractional Schrödinger system will be further discussed. The project has a wide physical meaning and is a further discussion on some existing important results about the nonlinear Schrödinger system.

英文关键词： Schrödinger equation;critical exponent;variational method;existence of solutions;nondegeneracy