非线性微分差分方程组多重解的存在性

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摘要

非线性微分差分方程组是现代数学的一个重要分支，无论在理论中还是在实

际应用中，非线性微分差分方程组均被用来描述力学、控制过程、生态与经济系

统及流行病等领域的问题．很多的实际问题最终都化为求解非线性微分差分方程

组，而非线性微分差分方程组解的存在性研究具有较长的历史，在此基础上也产

生了很多新的理论和研究成果，由于其应用的广泛性，非线性微分差分方程组解

的存在性还有很多需要研究和完善的工作，而且随着过程的发展会不断产生许多

新的数学方法和结论

本文主要研究了一类带有(p,q)-Laplacian 算子的拟线性椭圆型偏微分方程组

Dirichelet 问题的非平凡解与无穷多非平凡解的存在性，以及讨论了 2n 阶非线性差

分方程组在超线性、渐近线性情形下周期解的存在性结果．

第一章绪论，论述了本文的变分背景知识，总结了近年来国内外许多学者在

该研究领域的研究进展和主要成果．

第二章给出了本文所需要用到的定义、定理、引理等基础知识．

第三章，利用变分法中的局部环绕定理讨论含(p,q)-Laplacian 算子的椭圆型方

程组

(P)

















,,0

,),,,(),,()(

xvu

xvuxFvuxGuudivv

xvuxFvuxGuudivu



多重解的存在性问题.局部环绕定理是环绕定理的推广和应用，而且在解决非线性

微分方程组非平凡多解的情况下,与环绕定理相比应用更为广泛，同时也是证明非

线性方程组多重解的重要工具.

第四章，利用临界点理论中的环绕定理和鞍点定理，证明了一类 2n 阶非线性

差分方程组

(Q)

















,,0),()1()(

zkykgyr

zkxkfxr

在超线性及渐近线性情形下周期解的存在性．

第五章是总结与展望，综述了本文的主要工作，并对进一步的深入研究提出

了展望．

关键词:拟线性椭圆型方程组局部环绕定理非线性差分方程组解

的存在性

ABSTRACT

Nonlinear differential difference equations is an important branch of modern

mathematics,Whether in theory or in practice,Nonlinear differential difference equations

are used to describe the problems,Which contains mechanics,control process,ecological

and ecomomic systems,chemical recycling and epidemiology,etc.But many practical

problems ultimately translate into the solution of the nonlinear differential difference

equations,Research of existence of solutions of the nonlinear differential difference

equations has a long history,there are many research methods about these problems.On

this basis,resulting in a lot of new theories and research findings,Because of its wide

application,there are a lot of work needed to study and improve existence of solutions of

the nonlinear differential difference equations.And with the development process,that

will continue to produce many new mathematical methods and theories.

In this paper, we discussed the dirichlet problem of a type of elliptic systems

with(p,q)-Laplacian operator,We obtain the existence of nontrivial solutions and

multiple solutions.Discussing the periodic problem of a class of 2n-order nonlinear

differential equations and proving the existence of periodic solutions under the

asymptoticlinear and superlinear cases respectively.

In Chapter 1, we introduced the background of the knowledge in variational area.We

also summarized the works and progress from many researchers in this field in recent

years.

In Chapter 2, we give some definitions, theorems and lemmas.

In Chapter 3, we use local linking theorem in critical point theory to discuss the

following Drichlet problem:

(P)

















,,0

,),,,(),,()(

xvu

xvuxFvuxGuudivv

xvuxFvuxGuudivu



and obtain the existence of multiple solutions. Local linking theorem is promotion and

application of linking theorem. Moreover, compared with the linking theorem, it applied

more widely, meanwhile, it is also proved an important tool for multiple solutions.

In Chapter 4, we use Linking theorem and Saddle point theorem in critical point

theory to discuss the 2n-order nonlinear diffenential equations:

(Q)

















,,0),()1()(

zkykgyr

zkxkfxr

We prove the existence of periodic solutions under the asymptoticlinear and superlinear

cases respectively.

In Chapter 5, we summarized our research in this article. We also gave our direction

for further research in this chapter.

Key word: Qusilinear elliptic systems, Local linking theorem,

Nonlinear differential equations, Existence of solutions

中文摘要

ABSTRACT

第一章绪论............................................................................................................... 1

§1.1 问题的意义和发展过程................................................................................. 1

§1.2 本文的主要研究内容....................................................................................... 2

第二章预备知识........................................................................................................... 4

§2.1 Sobolev 空间................................................................................................... 4

§2.2 弱收敛与弱紧性............................................................................................. 6

§2.3 极值原理与局部环绕定理............................................................................... 8

第三章一类(p,q)-Laplace 方程组多重解的存在性...................................................11

§3.1 引言............................................................................................................... 11

§3.2 主要引理......................................................................................................... 12

§3.3 特征值问题..................................................................................................... 13

§3.4 方程组多重解的存在性证明......................................................................... 14

§3.5 例子................................................................................................................. 20

第四章 2n 阶非线性差分方程组的周期解................................................................ 21

§4.1 引言............................................................................................................... 21

§4.2 主要定理......................................................................................................... 22

§4.3 超线性情形下的无穷多周期解..................................................................... 24

§4.4 渐近线性情形下的无穷多周期解................................................................. 26

第五章总结和展望..................................................................................................... 30

参考文献......................................................................................................................... 31

在读期间公开发表的论文和承担科研项目及取得成果............................................. 33

致谢........................................................................................................................... 34

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摘要：

摘要非线性微分差分方程组是现代数学的一个重要分支，无论在理论中还是在实际应用中，非线性微分差分方程组均被用来描述力学、控制过程、生态与经济系统及流行病等领域的问题．很多的实际问题最终都化为求解非线性微分差分方程组，而非线性微分差分方程组解的存在性研究具有较长的历史，在此基础上也产生了很多新的理论和研究成果，由于其应用的广泛性，非线性微分差分方程组解的存在性还有很多需要研究和完善的工作，而且随着过程的发展会不断产生许多新的数学方法和结论本文主要研究了一类带有(p,q)-Laplacian算子的拟线性椭圆型偏微分方程组Dirichelet问题的非平凡解与无穷多非平凡解的存在性，以及讨论了2n阶非线...

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