几类分数阶薛定谔方程多解的存在性

发布时间：2018-04-19 00:09

本文选题：分数阶薛定谔方程 + 对称的山路定理　；参考：《山东师范大学》2016年硕士论文

【摘要】：随着科学技术和现代数学基础理论的不断发展,出现的各种各样的非线性问题也日益引起人们的广泛重视,非线性泛函分析已成为现代数学的重要研究方向之一.非线性泛函分析又是非线性分析中的一个重要分支,它以数学和物理学中出现的非线性问题为背景,建立起了处理非线性问题的若干方法和理论.因其能很好的解释自然界各种现象而受到国内外数学界和自然科学界的重视.而非线性薛定谔方程来源于应用数学,物理学等各种应用学科,更是非线性微分方程中最活跃的领域之一.近年来,人们对薛定谔方程解的存在性得到了一些新的成果,而分数阶薛定谔方程多解的存在性问题又是近年来讨论的热点.本文主要利用变形的对称山路定理Nehari流形等临界点理论讨论了几类特殊的分数阶薛定谔方程多解的存在性的情况,并证明了其多解的存在性.本文分为以下三章：第一章是绪论,主要介绍了分数阶薛定谔方程的有关研究背景及相对应的空问和范数等有关知识.第二章研究了没有(A-R)条件的分数阶薛定谔方程：其中表示分数阶的拉普拉斯算子指数为α,f(x,u)是定义在RN×R上的连续函数,势函数V(x)是RN上的连续函数.我们利用变形的对称山路定理,在适当的条件下,证明其无穷多个解的存在性.第三章研究了势函数为无界函数的分数阶薛定谔方程：C(RN)且在RN上变号.无界势函数V(x)在RN上连续.我们利用Nehari流形,在适当的条件下,证明其多个解的存在性.
[Abstract]:With the development of science and technology and the basic theory of modern mathematics, people pay more and more attention to all kinds of nonlinear problems, and nonlinear functional analysis has become one of the important research directions in modern mathematics.Nonlinear functional analysis is also an important branch of nonlinear analysis. Based on nonlinear problems in mathematics and physics, some methods and theories for dealing with nonlinear problems are established.Because of its ability to explain all kinds of phenomena in nature, it has been paid attention to by the mathematics and natural sciences at home and abroad.The nonlinear Schrodinger equation comes from applied mathematics and physics and is one of the most active fields of nonlinear differential equations.In recent years, some new results have been obtained on the existence of solutions for Schrodinger equation, and the existence of multiple solutions for fractional Schrodinger equation is a hot topic in recent years.In this paper, the existence of multiple solutions for some special fractional Schrodinger equations is discussed by using the critical point theory of the Nehari manifold of the symmetric mountain path theorem of deformation, and the existence of the multiple solutions is proved.This paper is divided into three chapters: the first chapter is an introduction, mainly introduces the research background of fractional Schrodinger equation and the corresponding knowledge of space and norm.In chapter 2, we study the fractional Schrodinger equation without the A-R condition, where the Laplace operator exponent representing the fractional order is defined as a continuous function on RN 脳 R, and the potential function VX is a continuous function on RN.By using the symmetric mountain path theorem of deformation, we prove the existence of infinite solutions under appropriate conditions.In chapter 3, we study the fractional Schrodinger equation with unbounded potential function: C _ n) and change the sign on RN.The unbounded potential function VX) is continuous on RN.In this paper, we prove the existence of several solutions of Nehari manifolds under proper conditions.
【学位授予单位】：山东师范大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O175

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