【摘要】：從二十世紀初開始,微分方程邊值問題逐漸成為了微分方程研究中的熱門問題,特別是Dirichlet、Neumann等邊值問題解的存在性以及多解性。數(shù)年來,由于在物理學、航天、生物學等領(lǐng)域微分方程邊值問題的廣泛應用,很多關(guān)于線性、擬線性橢圓方程的各類邊值問題已經(jīng)有了比較豐富的結(jié)果。在此背景下,雙調(diào)和及p-雙調(diào)和微分方程邊值問題激發(fā)了許多學者的研究熱情,并取得了較為顯著的成果。但由于p(x)-biharmonic算子具有相對復雜的非線性性質(zhì),許多經(jīng)典理論和方法都無法使用,目前涉及p(x)-雙調(diào)和微分方程Neumann邊值問題的內(nèi)容比較有限,因此,對此類微分方程邊值問題的研究具有十分重要的實際意義。本文主要運用變分法以及不同類型的臨界點定理,分別研究了帶有p(x)-biharmonic算子的具有連續(xù)非線性項、不連續(xù)非線性項的Neumann邊值問題是否有解以及解的個數(shù)的情況,得到了一些新的結(jié)果。第一章,首先介紹了研究帶有p(x)-biharmonic算子的Neumann邊值問題的背景、意義以及常見的研究此類微分方程解的存在性的方法,其次回顧了四階微分方程非線性邊值問題的研究背景及研究近況,最后對本文的核心研究內(nèi)容進行了描述。第二章,介紹了研究本文問題所涉及的臨界點定理。第三章,利用變分法、Ricceri三臨界點定理研究了帶有p(x)-雙調(diào)和算子的具有連續(xù)非線性項的Neumann邊值問題解的存在性,獲得了所研究問題至少具有三個解的存在性結(jié)論。第四章,利用變分法、非光滑臨界點定理研究了帶有p(x)-雙調(diào)和算子的具有不連續(xù)非線性項的Neumann邊值問題解的存在性,獲得了所研究問題存在一個解的結(jié)論。第五章,總結(jié)了本文的研究內(nèi)容以及主要的研究成果,并對此后的研究內(nèi)容進行了瞻望預期。
[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.
【學位授予單位】：北京郵電大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：O175.8

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