四阶微分方程解的存在性研究
发布时间：2018-09-18 08:09
【摘要】：从二十世纪初开始,微分方程边值问题逐渐成为了微分方程研究中的热门问题,特别是Dirichlet、Neumann等边值问题解的存在性以及多解性。数年来,由于在物理学、航天、生物学等领域微分方程边值问题的广泛应用,很多关于线性、拟线性椭圆方程的各类边值问题已经有了比较丰富的结果。在此背景下,双调和及p-双调和微分方程边值问题激发了许多学者的研究热情,并取得了较为显著的成果。但由于p(x)-biharmonic算子具有相对复杂的非线性性质,许多经典理论和方法都无法使用,目前涉及p(x)-双调和微分方程Neumann边值问题的内容比较有限,因此,对此类微分方程边值问题的研究具有十分重要的实际意义。本文主要运用变分法以及不同类型的临界点定理,分别研究了带有p(x)-biharmonic算子的具有连续非线性项、不连续非线性项的Neumann边值问题是否有解以及解的个数的情况,得到了一些新的结果。第一章,首先介绍了研究带有p(x)-biharmonic算子的Neumann边值问题的背景、意义以及常见的研究此类微分方程解的存在性的方法,其次回顾了四阶微分方程非线性边值问题的研究背景及研究近况,最后对本文的核心研究内容进行了描述。第二章,介绍了研究本文问题所涉及的临界点定理。第三章,利用变分法、Ricceri三临界点定理研究了带有p(x)-双调和算子的具有连续非线性项的Neumann边值问题解的存在性,获得了所研究问题至少具有三个解的存在性结论。第四章,利用变分法、非光滑临界点定理研究了带有p(x)-双调和算子的具有不连续非线性项的Neumann边值问题解的存在性,获得了所研究问题存在一个解的结论。第五章,总结了本文的研究内容以及主要的研究成果,并对此后的研究内容进行了瞻望预期。
[Abstract]:Since the beginning of the 20th century, the boundary value problem of differential equations has gradually become a hot problem in the study of differential equations, especially the existence and multiple solutions of solutions for Dirichlet,Neumann equilateral value problems. In recent years, due to the wide application of boundary value problems of differential equations in physics, aerospace, biology and other fields, a lot of results have been obtained for various boundary value problems of linear and quasilinear elliptic equations. In this context, the boundary value problems of biharmonic and p-biharmonic differential equations have aroused the enthusiasm of many scholars, and have achieved remarkable results. However, due to the relatively complex nonlinear properties of the p (x) biharmonic operator, many classical theories and methods cannot be used. At present, the contents of Neumann boundary value problems for p (x) biharmonic differential equations are limited. The study of boundary value problems for this kind of differential equations is of great practical significance. In this paper, the variational method and the critical point theorems of different types are used to study the existence of solutions and the number of solutions for Neumann boundary value problems with continuous nonlinear terms and discontinuous nonlinear terms with p (x)-biharmonic operators, respectively. Some new results have been obtained. In the first chapter, the background and significance of the study of Neumann boundary value problems with p (x) harmonic operator are introduced, and the common methods to study the existence of solutions of this kind of differential equations are discussed. Secondly, the research background and recent situation of nonlinear boundary value problems for fourth order differential equations are reviewed. Finally, the core contents of this paper are described. In the second chapter, the critical point theorem is introduced. In chapter 3, the existence of solutions for Neumann boundary value problems with continuous nonlinear terms with p (x) biharmonic operators is studied by using the variational method Ricceri triple critical point theorem, and the existence of at least three solutions is obtained. In chapter 4, by using variational method and nonsmooth critical point theorem, we study the existence of solutions for Neumann boundary value problems with discontinuous nonlinear terms with p (x) biharmonic operators, and obtain a conclusion that there exists a solution to the problem studied. The fifth chapter summarizes the research content and the main research results, and looks forward to the future research content.
【学位授予单位】：北京邮电大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.8
本文编号：2247296
[Abstract]:Since the beginning of the 20th century, the boundary value problem of differential equations has gradually become a hot problem in the study of differential equations, especially the existence and multiple solutions of solutions for Dirichlet,Neumann equilateral value problems. In recent years, due to the wide application of boundary value problems of differential equations in physics, aerospace, biology and other fields, a lot of results have been obtained for various boundary value problems of linear and quasilinear elliptic equations. In this context, the boundary value problems of biharmonic and p-biharmonic differential equations have aroused the enthusiasm of many scholars, and have achieved remarkable results. However, due to the relatively complex nonlinear properties of the p (x) biharmonic operator, many classical theories and methods cannot be used. At present, the contents of Neumann boundary value problems for p (x) biharmonic differential equations are limited. The study of boundary value problems for this kind of differential equations is of great practical significance. In this paper, the variational method and the critical point theorems of different types are used to study the existence of solutions and the number of solutions for Neumann boundary value problems with continuous nonlinear terms and discontinuous nonlinear terms with p (x)-biharmonic operators, respectively. Some new results have been obtained. In the first chapter, the background and significance of the study of Neumann boundary value problems with p (x) harmonic operator are introduced, and the common methods to study the existence of solutions of this kind of differential equations are discussed. Secondly, the research background and recent situation of nonlinear boundary value problems for fourth order differential equations are reviewed. Finally, the core contents of this paper are described. In the second chapter, the critical point theorem is introduced. In chapter 3, the existence of solutions for Neumann boundary value problems with continuous nonlinear terms with p (x) biharmonic operators is studied by using the variational method Ricceri triple critical point theorem, and the existence of at least three solutions is obtained. In chapter 4, by using variational method and nonsmooth critical point theorem, we study the existence of solutions for Neumann boundary value problems with discontinuous nonlinear terms with p (x) biharmonic operators, and obtain a conclusion that there exists a solution to the problem studied. The fifth chapter summarizes the research content and the main research results, and looks forward to the future research content.
【学位授予单位】：北京邮电大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.8
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