一些复杂非线性椭圆问题的研究

发布时间：2018-01-29 08:04

本文关键词： 变分方法山路引理 Ljusternik-Schnirelmann型极小极大原理流不变集亏格 Musielak-Orlicz-Sobolev空间 Brouwer不动点定理 Lyapunov-Schmidt约化 Pohozaev流形　出处：《兰州大学》2016年博士论文　论文类型：学位论文

【摘要】：本博士学位论文主要考虑了几类具有复杂非线性项的椭圆问题解的存在性及多解性.在这里,复杂非线性是指：带有非线性边界条件,算子是非线性的,外力项不满足(AR)条件.本文共有六章：第一章是绪论,主要介绍了本文的研究背景,研究的问题以及得到的主要结论.第二章是预备知识,介绍了后面要用到的比较原理,非线性泛函分析知识,变分法以及椭圆方程的一些重要的结论.第三章,我们研究了有界区域上的带有非线性边界条件的椭圆方程其中Ω(?)RN是有界光滑区域,N≥3.在本章的第一节中,我们讨论非线性项f,g满足渐近线性条件时,多重变号解的存在性.在第二节中,我们考虑非线性项是凹凸非线性时无穷多个变号解的存在性.研究方法是结合流不变集和Ljusternik-Schnirelman型极小极大原理.第四章,我们将考虑Musielak-Orlicz-Sobolev空间中椭圆问题的弱解的存在性和多重性.其中Ω是RN中的有界光滑区域,n才表示(?)Ω的外法向量.通过一个最新的研究结果Musielak-Orlicz-Sobolev空间中的边界上的紧嵌入定理,我们得以研究Musielak-Orlicz-Sobolev空间中带有非线性边界条件的拟线性椭圆问题解的存在性和多解性.第五章,我们主要考虑下面的椭圆问题的单峰正解和多峰正解的存在性：其中表示对应于(?)R+N型的外法向导数；利用Brouwer不动点定理和Lyapunov-Schmidt约化过程得到了问题的单峰解和多峰解.第六章,我们考虑了一类Schrodinger方程的多解性.这里关于非线性项f(u)的条件是一般性的,通过对系数α(x)的要求,利用Po一hozaev流形中的子流形,我们分别得到了定号解和变号解的存在性.
[Abstract]:In this dissertation, we mainly consider the existence and multiple solutions of solutions for several kinds of elliptic problems with complex nonlinear terms. In this case, complex nonlinearity means that operators are nonlinear with nonlinear boundary conditions. There are six chapters in this paper: the first chapter is the introduction, which mainly introduces the research background, the research problems and the main conclusions. The second chapter is the preparatory knowledge. The principle of comparison, the knowledge of nonlinear functional analysis, the variational method and some important conclusions of elliptic equations are introduced. Chapter 3. In this paper, we study the elliptic equations with nonlinear boundary conditions in a bounded domain where 惟? In the first section of this chapter, we discuss the existence of multiple sign solutions when the nonlinear term fg satisfies the asymptote linear condition. We consider the existence of infinitely many sign solutions when the nonlinear term is concave and convex nonlinear. The method is to combine the flow invariant set with the Ljusternik-Schnirelman type minimax principle. Four chapters. We will consider the existence and multiplicity of weak solutions for elliptic problems in Musielak-Orlicz-Sobolev spaces, where 惟 is the bounded smooth domain n in RN. By a new research result, the compact embedding theorem on the boundary of Musielak-Orlicz-Sobolev space. We study the existence and multiplicity of solutions for quasilinear elliptic problems with nonlinear boundary conditions in Musielak-Orlicz-Sobolev spaces. Chapter 5th. We mainly consider the existence of single-peak positive solutions and multi-peak positive solutions for the following elliptic problems: where the representation corresponds to? The external guide number of R N type; By using Brouwer fixed point theorem and Lyapunov-Schmidt reduction process, the single-peak solution and multi-peak solution of the problem are obtained. Chapter 6th. We consider the multiplicity of solutions for a class of Schrodinger equations, where the conditions for the nonlinear term f u) are general and require the coefficient 伪 x). By using submanifolds in Po-hozaev manifolds, we obtain the existence of definite solutions and variant solutions, respectively.
【学位授予单位】：兰州大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175.25

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