关于几类非线性问题的单号解和变号解研究

发布时间：2018-05-16 11:54

本文选题：变分方法 + 临界点定理　；参考：《云南大学》2016年博士论文

【摘要】：本文主要使用变分方法和临界点理论研究几类非线性问题山路型单号解和变号解的存在性和多重性.全文共分为四章,第一章介绍一些背景知识和本文的研究内容,其余各章的主要内容分别为：第二章研究非线性Kirchhoff型方程非线性Kirchhoff型方程一直备受关注,在许多领域都有广泛的应用,据我们所知,现有的文献都是在有界区域Ω上研究其变号解.有别于此,我们在R3上研究该方程山路型正解、负解和变号解的存在性及无穷多山路型变号解的存在性.第三章研究非线性Schrodinger方程山路型单号解和变号解的存在性和多重性,当非线性项h满足适当的增长性条件和全局(AR)条件时,文献[22]研究了其变号解的存在性；文献[105]研究了其正解、负解和无穷多变号解的存在性.相异于文献[22]和[105],我们在局部(AR)条件和较弱的条件下研究了该方程山路型正解、负解和变号解的存在性及无穷多山路型变号解的存在性.这一章的结果改进了文献[22]和[105]中的结果.第四章分别研究拟线性椭圆方程拟线性Schrodinger方程和广义拟线性Schrodinger方程山路型正解、负解、变号解及无穷多变号解的存在性.由于拟线性项的存在使得上述问题的相应的能量泛函在其熟知的工作空间上没有定义,我们不能直接对其使用变分方法.为克服这一困难,我们首先使用变量替换的方法将其转化为半线性问题,然后以熟知的Sobolev空间作为相应变更泛函的工作空间,在适当的条件下得到其山路型正解、负解和变号解的存在性.进而,在非线性项h(x,t)关于第二变元是奇函数的情形下,得到其无穷多山路型变号解的存在性.
[Abstract]:In this paper, variational method and critical point theory are used to study the existence and multiplicity of single-sign solutions and variable-sign solutions for some nonlinear problems. This paper is divided into four chapters. The first chapter introduces some background knowledge and the main contents of this paper. The main contents of the other chapters are as follows: in chapter two, the study of nonlinear Kirchhoff equation of nonlinear Kirchhoff type equation has been paid more and more attention. It has been widely used in many fields. As far as we know, the existing literature is to study its sign solution on bounded domain 惟. Different from this, we study the existence of positive solution, negative solution and sign solution of the equation on R3, and the existence of infinite mountain path variant solution. In chapter 3, we study the existence and multiplicity of single-sign solutions and variable-sign solutions of nonlinear Schrodinger equation. When the nonlinear term h satisfies the appropriate growth condition and global AR-condition, the existence of the sign variation solution is studied in [22]. In reference [105], the existence of positive solutions, negative solutions and infinitely variable sign solutions was studied. In contrast to references [22] and [105], we study the existence of positive, negative and sign solutions and the existence of infinite mountain path variant solutions for the equation under local and weak ARs conditions. The results of this chapter improve the results in references [22] and [105]. In chapter 4, we study the existence of mountain path solutions, negative solutions, variable sign solutions and infinitely variable sign solutions for quasilinear elliptic equations with quasilinear Schrodinger equations and generalized quasilinear Schrodinger equations, respectively. Due to the existence of the quasilinear term, the corresponding energy functional of the above problem is not defined in its familiar workspace, so we can not directly use the variational method. In order to overcome this difficulty, we first use the method of variable substitution to transform it into a semilinear problem, and then take the familiar Sobolev space as the workspace of the corresponding change functional, and obtain its mountain path positive solution under appropriate conditions. Existence of negative solution and sign change solution. Furthermore, in the case that the second variable is an odd function, the existence of an infinite number of mountain path variant solutions is obtained.
【学位授予单位】：云南大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175

【相似文献】