【作者】 刘芳；

【导师】 代国伟；

【作者基本信息】 大连理工大学 ， 基础数学， 2023， 博士

【摘要】 分歧理论是非线性泛函分析的重要组成部分.本文利用分歧方法研究了两类椭圆边值问题解的存在性:一类是超定边值问题,主要寻找解及其有解区域;一类是Kirchhoff型问题,主要建立其单边全局分歧结构,并由此确定相关Kirchhoff型问题结点解存在的参数范围.本文主要内容如下:第1部分简要介绍了两类问题的研究背景以及国内外研究现状,同时给出了研究这些问题所需要的一些基础知识.第2部分研究了Serrin超定问题的局部和全局分歧结构.本文证明了局部分歧曲线是临界的,即参数在分歧点处的导数等于零.为了得到全局分歧结构,建立了一个有限维版本的全局分歧定理.第3部分考虑了与带电液滴模型相关的超定椭圆边值问题.借助于CrandallRabinowitz局部分歧定理,本文证明了存在从外域分歧出一族非平凡无界区域,使得该超定问题在其上有正的有界解.第4部分考虑了带Schr?dinger项的超定椭圆边值问题.基于Crandall-Rabinowitz局部分歧定理,本文分别获得了从外域和直圆柱的补域分歧出的一族非平凡的无界域使得该超定问题在其上存在有界正解.直圆柱的补域方面的结论可以看作BCN(BerestyckiCaffarelli-Nirenberg)猜想在该类区域上的一个否定回答.Ros、Ruiz和Sicbaldi于2020年发表在JEMS上的文章是基于Krasnosel’skii局部分歧定理而获得的一列非平凡外区域.本文得到了一族非平凡外区域,且推广到所有特征值.第5部分考虑了Kirchhoff型问题的单边全局分歧现象,并研究了其结点解集分支的全局结构.为此,完整刻画了一个齐次非局部特征值问题的谱.第6部分做了总结及展望.更多还原

【Abstract】 The bifurcation theory is one important part of nonlinear functional analysis.In this article,the existence of solutions for two classes of elliptic boundary value problems are studied by means of bifurcation theory.One class is the overdetermined boundary value problems,whose solutions and domains are primarily sought.The other class is the Kirchhoff problems,whose unilateral global bifurcation structure is mainly established.Furthermore,the parameter range of the node solution to a Kirchhoff type problem is determined.The framework of this article is as follows:In Chapter 1,the background of the two classes of problems,the research history and the basic theories are briefly introduced.In Chapter 2,the local and global bifurcation structure of the Serrin overdetermined problem are studied.Here the local bifurcation curve is critical,i.e.the derivative of the parameter at the bifurcation point is equal to zero.In order to obtain the global bifurcation structure,a finite dimensional version of the global bifurcation theorem is established.In Chapter 3,an overdetermined elliptic boundary value problem related to the charged droplet model is considered.With the help of the Crandall-Rabinowitz local bifurcation theorem,a family of nontrivial unbounded domains bifurcating from the exterior domains is determined,which ensures that the overdetermined problem has a positive bounded solution.In Chapter 4,an overdetermined elliptic boundary value problem with a Schr ?dinger term is considered.Based on the Crandall-Rabinowitz local bifurcation theorem,a family of nontrivial unbounded domains is obtained bifurcating from the exterior domains and the complement domains of the straight cylinder respectively,on which the overdetermined problem has a bounded positive solution.The conclusion on the complement domains of a straight cylinder can be regarded as a negative answer to the BCN(Berestycki-Caffarelli-Nirenberg)conjecture.A paper has been published in JEMS in 2020 by Ros,Ruiz and Sicbaldi.They are a sequence of nontrivial exterior domains based on the Krasnosel’skii local bifurcation theorem.In this paper,a family of nontrivial exterior domains is obtained and extended to the cases of all eigenvalues.In Chapter 5,the unilateral global bifurcation of a Kirchhoff type problem and the global structure of the branch of its nodal solution set are considered.To this end,the spectra of a homogeneous nonlocal eigenvalue problem are described.In Chapter 6,we make a summary and prospection to the future work.更多还原