發(fā)布時(shí)間：2018-04-19 00:09

  本文選題：分?jǐn)?shù)階薛定諤方程 + 對(duì)稱的山路定理　； 參考：《山東師范大學(xué)》2016年碩士論文

【摘要】：隨著科學(xué)技術(shù)和現(xiàn)代數(shù)學(xué)基礎(chǔ)理論的不斷發(fā)展,出現(xiàn)的各種各樣的非線性問題也日益引起人們的廣泛重視,非線性泛函分析已成為現(xiàn)代數(shù)學(xué)的重要研究方向之一.非線性泛函分析又是非線性分析中的一個(gè)重要分支,它以數(shù)學(xué)和物理學(xué)中出現(xiàn)的非線性問題為背景,建立起了處理非線性問題的若干方法和理論.因其能很好的解釋自然界各種現(xiàn)象而受到國(guó)內(nèi)外數(shù)學(xué)界和自然科學(xué)界的重視.而非線性薛定諤方程來源于應(yīng)用數(shù)學(xué),物理學(xué)等各種應(yīng)用學(xué)科,更是非線性微分方程中最活躍的領(lǐng)域之一.近年來,人們對(duì)薛定諤方程解的存在性得到了一些新的成果,而分?jǐn)?shù)階薛定諤方程多解的存在性問題又是近年來討論的熱點(diǎn).本文主要利用變形的對(duì)稱山路定理Nehari流形等臨界點(diǎn)理論討論了幾類特殊的分?jǐn)?shù)階薛定諤方程多解的存在性的情況,并證明了其多解的存在性.本文分為以下三章：第一章是緒論,主要介紹了分?jǐn)?shù)階薛定諤方程的有關(guān)研究背景及相對(duì)應(yīng)的空問和范數(shù)等有關(guān)知識(shí).第二章研究了沒有(A-R)條件的分?jǐn)?shù)階薛定諤方程：其中表示分?jǐn)?shù)階的拉普拉斯算子指數(shù)為α,f(x,u)是定義在RN×R上的連續(xù)函數(shù),勢(shì)函數(shù)V(x)是RN上的連續(xù)函數(shù).我們利用變形的對(duì)稱山路定理,在適當(dāng)?shù)臈l件下,證明其無窮多個(gè)解的存在性.第三章研究了勢(shì)函數(shù)為無界函數(shù)的分?jǐn)?shù)階薛定諤方程：C(RN)且在RN上變號(hào).無界勢(shì)函數(shù)V(x)在RN上連續(xù).我們利用Nehari流形,在適當(dāng)?shù)臈l件下,證明其多個(gè)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.
【學(xué)位授予單位】：山東師范大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2016
【分類號(hào)】：O175

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本文編號(hào)：1770695