几类全空间上半线性椭圆型方程（组）解的存在性

发布时间：2018-05-11 22:10

本文选题：变分法 + Nehari流形　；参考：《东南大学》2017年博士论文

【摘要】：本文首先研究带凹凸非线性项Schrodinger方程和Kirchhoff方程在全空间中的多解性以及基态解的存在性,其次考虑Kirchhoff系统和带饱和非线性项Schrodinger系统的规范L2-解的存在性.主要内容安排如下:第一章大致回顾变分法的研究历程,介绍本文所涉及的临界点理论的相关知识,并对本文的主要工作和创新点进行概述.第二章考虑非线性项是次线性和超线性组合形式的Schrodinger方程,其中的位势函数有界,利用Nehari流形分块方法得到了方程的多解性和基态解的存在性.此结果推广了已有文献中位势恒为正常数非线性项凹凸的Schrodinger方程.第三章研究带凹凸非线性项Kirchhoff方程,在全空间上分别考虑了凸项为超三次次幂情形、凸项为次幂超线性项和凸项为一般超线性项情形,在适当的条件下得到了方程至少有两个解.第四章研究Kirchhoff系统,在不同的位势条件下分别考虑了 L2-次临界和L2-临界情形,证明了系统规范L2-解的存在性.把单个Kirchhoff方程L2-解的结果推广到了Kirchhoff系统.此外,我们考虑了耦合常数小于零时Kirchhoff系统L2-解的存在性.第五章在更加一般的限制条件下考虑带饱和非线性项Schrodinger系统,利用单个方程L2-解的性质排除半平凡解,进而得到系统的规范L2-解的存在性.此外,我们可以考虑多个分量的带饱和非线性项Schrodinger系统和带平方根项Schrodinger系统的规范L2-解的存在性.第六章对文章进行了总结,对未来的研究进行了展望.
[Abstract]:In this paper, we first study the multiplicity of solutions and the existence of ground state solutions of Schrodinger equation and Kirchhoff equation with concave and convex nonlinear terms in the whole space, and then consider the existence of normal L2-solutions for Kirchhoff systems and Schrodinger systems with saturated nonlinear terms. The main contents are as follows: the first chapter reviews the research history of the variational method, introduces the relevant knowledge of the critical point theory involved in this paper, and summarizes the main work and innovation of this paper. In chapter 2, we consider that the nonlinear term is a sublinear and superlinear Schrodinger equation, in which the potential function is bounded. By using the Nehari manifold block method, we obtain the existence of multiple solutions and ground state solutions of the equation. The results extend the concave and convex Schrodinger equation with the potential constant as a normal number and nonlinear term in the literature. In chapter 3, the Kirchhoff equation with concave and convex nonlinear terms is studied. In the whole space, the convex term is superlinear term, the convex term is a general superlinear term, the convex term is a superlinear term, and the convex term is a superlinear term. At least two solutions to the equation are obtained under proper conditions. In chapter 4, we study the Kirchhoff system. We consider the L2-subcritical and L2-critical cases under different potential conditions, and prove the existence of the canonical L2-solution of the system. The result of L _ 2-solution of a single Kirchhoff equation is extended to Kirchhoff system. In addition, we consider the existence of L2-solution for Kirchhoff system with coupling constant less than 00:00. In chapter 5, we consider the Schrodinger system with saturated nonlinear term under more general conditions. By using the properties of L _ 2-solution of a single equation, we exclude the semi-trivial solution, and then obtain the existence of the normal L _ 2-solution of the system. In addition, we can consider the existence of canonical L2-solutions for Schrodinger systems with saturated nonlinear terms and Schrodinger systems with square root terms. The sixth chapter summarizes the article and looks forward to the future research.
【学位授予单位】：东南大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O175.25

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