椭圆型方程基态解和集中性等若干问题的研究
发布时间：2018-11-20 10:58
【摘要】：本文我们研究椭圆型方程基态解和集中性等若干问题。我们着重研究 Neumann 边值问题、分数阶 Schrodinger 方程、Schrodinger-Poisson 系统和Kirchhoff方程。主要内容安排如下:第一章:我们回顾一些记号和约定,并给出在后面章节中会用到的一些有用的初步结果。第二章:我们研究次线性Neumann问题。与之前关于Dirichlet问题的工作相比,由于相应的变分泛函没有下界,因此一些困难随之产生。我们证明该Neumann问题有无穷多小负能量解,这补充了 Parini和Weth在[93]中关于最小能量解的近期工作。第三章:我们研究非线性分数阶Schrodinger方程L2-标准化解的存在性、不存在性和质量集中。与之前关于Schrodinger方程的工作相比,由于分数阶Laplacian的非局部性,因此我们遇到一些新的挑战。我们先证明了分数阶Gagliardo-Nirenberg-Sobolev不等式的最佳嵌入常数可以由精确的形式表达出来,这提高了 [57, 58]的工作。做到了以上这些,我们然后建立该方程L2-标准化解的存在性和不存在性。最后通过利用一些精巧的能量估计,在某类势阱下我们给出质量临界情形时的L2-标准化解集中行为的详细分析。第四章:我们研究Schrodinger-Poisson系统。由于它的物理相关性,因此三维Schrodinger-Poisson系统已被广泛研究和充分了解。相反地,本文关注的二维Schrodinger-Poisson系统的信息要少很多。Cingolani和Weth在[36]中已观察到Schrodinger-Poisson系统的变分结构会在二维情形时出现本质差异,这导致其解集会有更丰富的结构。然而[36]的变分方法仅限于p≥4的情形,这排除了一些物理有关的指标。本章我们将去掉这个令人不愉快的限制,并在2 p 4的情形下使用一个不同的变分方法探索更复杂的底层泛函几何。第五章:我们研究一类Kirchhoff方程。在适当的假设下,通过变分方法我们证明该方程基态解的存在性。此外我们还调查基态解的集中现象。
[Abstract]:In this paper, we study some problems such as the ground state solution and the centralization of the elliptic equation. We focus on the Neumann boundary value problem, fractional Schrodinger equation, Schrodinger-Poisson system and Kirchhoff equation. The main contents are as follows: chapter 1: we review some notation and conventions and give some useful preliminary results that will be used in later chapters. Chapter 2: we study sublinear Neumann problem. Compared with the previous work on the Dirichlet problem, some difficulties arise because the corresponding variational functional has no lower bound. We prove that the Neumann problem has infinitely small negative energy solutions, which complements the recent work of Parini and Weth on the minimum energy solution in [93]. Chapter 3: we study the existence, nonexistence and mass concentration of L _ 2-standard solutions for nonlinear fractional Schrodinger equations. Compared with previous work on Schrodinger equations, we meet some new challenges due to the nonlocality of fractional Laplacian. We first prove that the best embedding constant of fractional order Gagliardo-Nirenberg-Sobolev inequality can be expressed in exact form, which improves the work of [57, 58]. Then we establish the existence and nonexistence of the L 2-standard solution of the equation. Finally, by using some subtle energy estimates, we give a detailed analysis of the concentration behavior of the L2-standard solution in the critical case of mass under a certain kind of potential well. Chapter 4: we study Schrodinger-Poisson system. Because of its physical correlation, 3D Schrodinger-Poisson system has been widely studied and fully understood. On the contrary, the information of two-dimensional Schrodinger-Poisson systems is much less concerned in this paper. In [36], Cingolani and Weth have observed that the variational structures of Schrodinger-Poisson systems are essentially different in two-dimensional cases, which leads to a richer structure of their solution sets. However, the variational method of [36] is limited to the case of p 鈮，
本文编号：2344720
[Abstract]:In this paper, we study some problems such as the ground state solution and the centralization of the elliptic equation. We focus on the Neumann boundary value problem, fractional Schrodinger equation, Schrodinger-Poisson system and Kirchhoff equation. The main contents are as follows: chapter 1: we review some notation and conventions and give some useful preliminary results that will be used in later chapters. Chapter 2: we study sublinear Neumann problem. Compared with the previous work on the Dirichlet problem, some difficulties arise because the corresponding variational functional has no lower bound. We prove that the Neumann problem has infinitely small negative energy solutions, which complements the recent work of Parini and Weth on the minimum energy solution in [93]. Chapter 3: we study the existence, nonexistence and mass concentration of L _ 2-standard solutions for nonlinear fractional Schrodinger equations. Compared with previous work on Schrodinger equations, we meet some new challenges due to the nonlocality of fractional Laplacian. We first prove that the best embedding constant of fractional order Gagliardo-Nirenberg-Sobolev inequality can be expressed in exact form, which improves the work of [57, 58]. Then we establish the existence and nonexistence of the L 2-standard solution of the equation. Finally, by using some subtle energy estimates, we give a detailed analysis of the concentration behavior of the L2-standard solution in the critical case of mass under a certain kind of potential well. Chapter 4: we study Schrodinger-Poisson system. Because of its physical correlation, 3D Schrodinger-Poisson system has been widely studied and fully understood. On the contrary, the information of two-dimensional Schrodinger-Poisson systems is much less concerned in this paper. In [36], Cingolani and Weth have observed that the variational structures of Schrodinger-Poisson systems are essentially different in two-dimensional cases, which leads to a richer structure of their solution sets. However, the variational method of [36] is limited to the case of p 鈮，
本文编号：2344720
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