薛定谔泊松方程与基尔霍夫方程解的存在性研究

发布时间：2018-03-14 12:07

本文选题：薛定谔泊松方程　切入点：基尔霍夫型方程　出处：《湖南大学》2015年博士论文　论文类型：学位论文

【摘要】：本文运用变分技巧研究了两类具有临界指数增长的非局部椭圆偏微分方程的正解.第一类是薛定谔泊松方程.这类方程来自于量子电动力学,半导体理论等领域,是用来描述粒子在空间和时间中的运动规律的模型.实际上,当描述多个粒子的融合以及相互作用的运动规律时,则引进一个相应的非线性扰动项.而当考虑粒子运动过程中伴随有电场产生时,通常引进一个泊松扰动项.当非线性项具有临界指数增长时,我们得到了这类方程正基态解,与半经典解的存在性结果.此外,我们也得到了当普朗克常数趋向于零时半经典解的集中性与指数衰减性.第二类是基尔霍夫方程.这类方程不仅刻画可伸缩绳横向振动的长度变化,而且在非牛顿力学、宇宙物理、血浆问题和弹性理论、人口动力学等诸多领域都有广泛应用.当非线性项具有一般的临界指数增长时,我们研究了正基态解的存在性以及正约束态解的多重性.全文共分为五章,其主要内容如下:第一章着重介绍所研究的问题的物理背景、发展现状以及最新进展,然后对本文的工作做简要的介绍,并给出了一些所需的预备知识.在第二章,利用一个抽象的临界点定理以及集中紧致原理讨论了具有临界指数增长的薛定谔泊松方程正基态解的存在性.由于我们没有给出任何紧性假设条件,我们必须细心地分析该方程的极限形式.首先我们利用集中紧致原理在一个自然压缩下的流形中证明该极限方程至少存在一个正基态解.然后分析PalaisSmale序列的性质,建立一个局部紧性分裂定理.最后结合一个抽象的临界点定理去得到原方程基态解的存在性.在第三章,在没有假设函数t→f(t)/t3单调的条件下,我们研究了一类一般的临界指数增长的薛定谔泊松方程半经典解的存在性.由于线性扰动项可能是强制的,标准的索布列夫空间已经不适合我们这类问题的讨论.我们首先利用削减方法去证明修正方程至少存在一个山路解.然后细心地分析极限方程的性质并且建立一个新的局部紧性分裂定理.该紧性定理能够帮助我们得到原始方程至少存在一个正约束态解的结果.利用最大值原理我们也研究了正约束解的指数衰减性.在第四章,在没有单调性假设条件下,我们研究了一般临界指数增长的基尔霍夫方程的正基态解的存在性.通过把能量泛函压缩在一个抽象的Pohozaev流形中,我们利用径向空间中的紧嵌入性质证明极小化序列的收敛性.利用Pohozaev等式我们证明了最小能量值能用山路能量值刻画.在第五章,我们讨论了具有临界指数增长的基尔霍夫方程半经典解的多重性以及集中性与指数衰减性.在一些适度假设条件下,我们得到了该方程正解的个数与扰动图形结构有关的结论.此外,我们也证明了该方程至少存在一个基态解.我们的主要证明思想是根据线性扰动函数的每一个全局极小值点,我们定义一个Nehari解流形,然后利用一个广义重心映射对Nehari解流形进行区分.最后在每一个子流形中,利用集中紧致原理与Ekeland变分原理去研究极小化序列的收敛性.
[Abstract]:This paper uses variational techniques to study two kinds of positive solutions of nonlocal elliptic partial differential equation with critical exponent. The first is the Schrodinger Poisson equation. This equation is derived from quantum electrodynamics, semiconductor theory and other fields, is used to describe the particle motion in space and time in the model. In fact, when the law of motion the fusion of multiple particles and describe the interaction, the introduction of a perturbation of the corresponding nonlinear term. But when considering the particle movement process with electric field, usually the introduction of a Poisson perturbation. When the nonlinear term is critical exponent, we got this kind of equation is the ground state solution, and half the classical existence results. In addition, we also obtained when the Planck constant tends to concentrate and index at zero thirty classical decay of solutions. The second category is the Kirchhoff equation. This equation is not Only describe the length change of transverse vibration of the telescopic rope, but also in non Newtonian mechanics, cosmic physics, plasma and elastic theory, population dynamics and other fields are widely used. When the nonlinear term critical exponent with general growth, we studied is the existence of ground state solutions and positive solutions of multiple constraints the full thesis is divided into five chapters, the main contents are as follows: the first chapter introduces the physical background of the problem, the development status and the latest progress, then the work of this paper is briefly introduced, and gives some preliminary knowledge required. In the second chapter, using an abstract critical point theorem and the concentration compactness principle is discussed with Schrodinger Poisson equation with critical exponent is the existence of ground state solutions. Because we have not given any compactness assumptions, we must carefully analyze the polar equation The limit form. We use the concentration compactness principle proves that the equation has at least one positive solution in a natural state under compression. Then the manifold properties of PalaisSmale sequence analysis, to establish a local compactness theorem. Finally, a division of abstract critical point theorem to obtain the existence of solutions of the original equation of state in the third chapter, without assuming the function of t to f (T) /t3 monotone conditions, we study a class of general critical exponent Schrodinger Poisson equation semi classical solution. The linear disturbance may be mandatory, standard cable Buliefu space is not suitable for our discussion of this problem we first use the method of cutting. To prove the correction equation has at least one mountain solution. And then carefully analysis of the nature of the limit equation and establish a new division of the local compactness theorem. The compactness theorem Can help us get the original existence of at least one positive solution of the constraint state. The results we also study the positive solution of a constrained exponential decay principle. In the fourth chapter, in the absence of monotonicity assumption is the ground state we study the existence of solution of Kirchhoff equation in general critical exponent through. The energy functional compression in an abstract Pohozaev manifold, we use compact embedding properties in radial space convergence of minimizing sequence. By using the Pohozaev equation we prove that the minimum energy value can be described by the mountain energy value. In the fifth chapter, we discuss the critical exponent of Kirchhoff equation with semi classical multiple solutions as well as the concentration and attenuation. In some appropriate conditions, we obtain the equation is the number of solutions and perturbation graph structure of the conclusions. In addition, We also prove the existence of at least one ground state solution. Our main idea is to point value according to every global minimal linear perturbation function, we define a Nehari solution manifold, then Nehari solution manifold can be distinguished by the use of a generalized barycentric mapping. Finally, in each sub manifold, with convergence centralized principle and Ekeland compact variational principle to study the minimal sequence.

【学位授予单位】：湖南大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：O175

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