全空间上椭圆方程（系统）在临界条件下解的存在性及解的性态

发布时间：2018-05-10 10:58

本文选题：椭圆方程 + 变分法　；参考：《中国矿业大学》2016年博士论文

【摘要】：变分法是非线性泛函分析中重要的基本方法之一.它的基本思想是把微分方程解的问题归结为相应泛函的临界点问题.本文利用变分法研究了几类非线性椭圆方程(系统)在临界情形下的解的存在性和解的性态.这几类方程(系统)在物理和力学中都有非常有意义的实际应用背景,因此,它们一直都是数学工作者关注的热点.本文在适当的假设条件下证明了全空间上临界情形下这几类椭圆方程(系统)解或多解的存在性,并且分析了解的衰减、渐近等性态,改进和推广了一些已有的结果.本文内容共分为五章.第一章绪论部分对变分法进行了简单的概括,同时介绍了变分法中一些基本的结果、几类椭圆方程的背景、研究现状以及本文的主要工作.本文从第二章到第五章分别对全空间上的四类椭圆方程(系统)进行了研究,以往工作对相应问题多是对单个方程或者是非线性次临界问题进行讨论.而本文所讨论的问题均为全空间上临界情形下,因此所取得成果改进和完善了目前已有结果.第二章研究全空间上扰动椭圆系统解的存在性.在已有的参考文献中,很多研究单个扰动薛定谔方程,得到了最小能量解,或者是在有界域上针对椭圆系统给出了次临界情形下给出了问题解的存在性.而本文第二章是在上述文献的基础上,研究得到了全空间上两类扰动椭圆系统正解的存在性.推广和改进了已有的工作.第三章利用山路定理主要研究临界情形下带有磁势及一般非线性项的薛定谔系统,得到了该系统在全空间上解的存在性.以往的工作多为研究单个方程在次临界情形下带有特殊非线性项解的存在性及其相应的性态,本文的结果对以往的工作是一个改进和推广.从而进一步丰富了以往结果.第四章运用扩展的Clark定理研究了非线性薛定谔泊松方程,得到了无穷多解的存在性以及解衰减的性态.已有文献中,多讨论次临界情形下解的存在性,而且对于解的衰减这一性态没有研究,本文不仅给出了系统无穷多解的存在性还给出了解的衰减这一重要性态.第五章研究在全空间上带有临界增长非线性项的Kirchhoff型问题解的存在性,得到了非平凡解的存在性同时给出了解的渐近性态.文中证明了能量泛函满足局部紧性条件即(PS)c条件.与已有的工作相比较,许多文献研究的问题多为次临界情形或者是有界区域上的情形.
[Abstract]:Variational method is one of the important basic methods in nonlinear functional analysis. Its basic idea is to reduce the solution of differential equation to the critical point problem of corresponding functional. In this paper, the existence and behavior of solutions for some nonlinear elliptic equations (systems) under critical conditions are studied by variational method. These equations (systems) have a very meaningful practical application background in physics and mechanics, so they have always been the focus of attention of mathematics workers. In this paper, we prove the existence of solutions or multiple solutions of these elliptic equations (systems) under the condition of proper assumptions, and analyze the attenuation and asymptotic isomorphism of the solutions, and improve and generalize some existing results. This paper is divided into five chapters. The first chapter gives a brief summary of the variational method, introduces some basic results of the variational method, the background of several kinds of elliptic equations, the present research situation and the main work of this paper. In this paper, four kinds of elliptic equations (systems) on the whole space are studied from the second to the fifth chapters. In the past, the corresponding problems are mostly discussed on the single equation or the nonlinear subcritical problem. The problems discussed in this paper are all in the critical case of the whole space, so the results obtained improve and perfect the existing results. In chapter 2, we study the existence of solutions for perturbed elliptic systems in all spaces. In previous references, we have studied the single perturbation Schrodinger equation and obtained the minimum energy solution, or we have given the existence of the solution for the elliptic system under the subcritical condition in the bounded domain. In the second chapter, we study the existence of positive solutions for two kinds of perturbed elliptic systems in the whole space. The existing work has been popularized and improved. In chapter 3, we mainly study the Schrodinger system with magnetic potential and general nonlinear terms in the critical case by using the mountain pass theorem, and obtain the existence of the solution of the system on the whole space. Most of the previous work is to study the existence and corresponding behavior of the solution of a single equation with a special nonlinear term in the subcritical case. The results in this paper are an improvement and generalization of the previous work. This further enriches the previous results. In chapter 4, we use the extended Clark theorem to study the nonlinear Schrodinger Poisson equation, and obtain the existence of infinite solutions and the behavior of solution decay. In the literature, the existence of solutions in the case of subcritical conditions is discussed, and there is no study on the decay of solutions. In this paper, not only the existence of infinite solutions of systems is given, but also the important state of attenuation of solutions is given. In chapter 5, we study the existence of solutions for Kirchhoff type problems with critical growth nonlinear terms in the whole space, and obtain the existence of nontrivial solutions and the asymptotic behavior of solutions. In this paper, it is proved that the energy functional satisfies the local compactness condition. Compared with the previous work, most of the problems studied in the literature are subcritical case or bounded region case.
【学位授予单位】：中国矿业大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175.25

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