具有非局部项的非线性椭圆方程的解
发布时间：2018-11-09 18:12
【摘要】：本文主要研究了两类带有非局部项的椭圆方程解的存在性问题.首先,我们研究了Kirchhoff类型椭圆方程解的存在性,其中包括基态解、多解、负能量解和变号解的存在性.其次,我们对Schr?dinger-Poisson方程组解的存在性问题进行了讨论.具体的结论与使用方法如下.本文共分为如下六个章节:第一章,绪论.主要讲述本文所考虑的两类带有非局部项椭圆方程的物理背景与学术意义以及目前国内外的发展现状,最后简单的介绍本文的主要结论和本文所使用的符号.第二章,考虑带有Sobolev临界指数的Kirchhoff类型的椭圆方程,在(4)N?N?中的有界区域内,我们讨论了几种解的存在性问题.当N?4时,在q?2和q?(2,4)的情况下,我们使用Nehari流形以及Lions的第一集中紧性原理,克服获得(PS)序列有界性的困难,再结合Lions的第二集中紧性原理获得了(PS)序列的局部紧性,最后获得方程基态解bu的存在性.我们并且考虑当b?0时,Kirchhoff类型椭圆方程与Brezis-Nirenberg问题的关系,获得了当b?0时,Kirchhoff类型椭圆方程的基态解bu收敛于Brezis-Nirenberg问题的基态解0u.当q?(1,2)时,使用一种截断技术截断能量泛函,再使用Krasnoselskii’s genus(2?指标)理论,获得方程有无穷多负能量解.当N?5时,对任意的q?(1,2*)和在a,b满足一些假设条件情况下,证明了方程存在正解.第三章,讨论带有纯幂次非线性项的Kirchhoff类型椭圆方程变号解的存在性.我们使用变分法与不变集的下降流以及2?指标理论,获得了当p?(2,6)时,方程存在无穷多高能量变号解.我们的这个工作扩展了现有文献仅在p?(4,6)时存在变号解结论.第四章,讨论带有一般非线性项的Kirchhoff类型椭圆方程变号解的存在性问题.通过对非线性项的一些较弱假设,运用一个截断技术,再使用变号的Nehari流形与集中紧性原理,最后证明方程存在一个最小能量变号解和一个不变号的基态解,且这个变号解的能量严格大于基态解能量.第五章,研究另外一种类型的带有非局项的椭圆方程,即带有Sobolev临界指数的Schr?dinger-Poisson方程组.首先由Lax-Milgram定理知Schr?dinger-Poisson方程组可以转化成为一个包含非局部项的单一方程,再使用变分法,获得能量泛函.当p?1时,把能量泛函约束在Nehari流形上,得到(PS)序列的有界性,再使用集中紧性原理获得(PS)序列的局部紧性,最后再证明了Nehari流形对能量泛函的约束是一个自然约束,因此,获得原方程组存在一个基态解.而当p?(0,1)时,运用一种截断技术截断能量泛函,再对获得的新能量泛函运用2?指标理论,我们证明了方程存在无穷多负能量解.第六章,我们简单地把本论文加以总结,同时希望对本文中的一些结论能进一步优化以及弱化一些假设.
[Abstract]:In this paper, we study the existence of solutions for two classes of elliptic equations with nonlocal terms. First, we study the existence of solutions for Kirchhoff type elliptic equations, including ground state solutions, multiple solutions, negative energy solutions and sign changing solutions. Secondly, we discuss the existence of solutions for Schr?dinger-Poisson equations. The specific conclusions and usage methods are as follows. This article is divided into the following six chapters: the first chapter, introduction. This paper mainly describes the physical background and academic significance of the two kinds of elliptic equations with nonlocal terms considered in this paper, as well as the present development situation at home and abroad. Finally, the main conclusions and symbols used in this paper are briefly introduced. In chapter 2, we consider the elliptic equation of Kirchhoff type with Sobolev critical exponent. In this paper, we discuss the existence of some solutions in the bounded domain of. When N4, we use the Nehari manifold and the first concentrated compactness principle of Lions to overcome the difficulty of obtaining the boundedness of (PS) sequences in the case of Q2 and Q2 (2? 4). The local compactness of (PS) sequences is obtained by combining the second set compactness principle of Lions. Finally, the existence of the ground state solution bu of the equation is obtained. We also consider the relation between the Kirchhoff type elliptic equation and the Brezis-Nirenberg problem when b0, and we obtain that the ground state solution of the Kirchhoff type elliptic equation converges to the ground state solution of the Brezis-Nirenberg problem when b0. When Q2, a truncation technique is used to truncate the energy functional, and then Krasnoselskii's genus (2? Index) theory, the equation has infinitely many negative energy solutions. The existence of positive solutions to the equation is proved when N5 satisfies some hypotheses for any Q? (1 ~ 2 *) and a b. In chapter 3, we discuss the existence of sign change solutions for Kirchhoff type elliptic equations with pure power nonlinear terms. We use the variational method and the descending flow of invariant sets and 2? Based on the index theory, it is obtained that the equation has infinitely high energy sign solutions when p _ (2 ~ (2) ~ (6). Our work extends the existing literature only when p? (4 ~ (6) has a sign solution. In chapter 4, we discuss the existence of sign variation solutions for Kirchhoff type elliptic equations with general nonlinear terms. By using a truncation technique and the Nehari manifold of sign variation and the principle of centralization compactness, it is proved that the equation has a minimum energy sign solution and a ground state solution of invariant sign. Moreover, the energy of the signed solution is strictly larger than that of the ground state solution. In chapter 5, we study another type of elliptic equations with nonlocal terms, that is, Schr?dinger-Poisson equations with Sobolev critical exponents. First, the Lax-Milgram theorem shows that the Schr?dinger-Poisson equations can be transformed into a single equation containing nonlocal terms, and then the energy functional is obtained by using the variational method. When p? 1, the energy functional is confined to the Nehari manifold, the boundedness of the (PS) sequence is obtained, and the local compactness of the (PS) sequence is obtained by using the centralization compactness principle. Finally, it is proved that the constraint of the Nehari manifold on the energy functional is a natural constraint. Therefore, the existence of a ground state solution for the original equations is obtained. When p? (0 ~ 1), a truncation technique is used to truncate the energy functional, and then 2? In the index theory, we prove that there are infinite negative energy solutions to the equation. In the sixth chapter, we summarize this paper briefly, and hope that some conclusions in this paper can be further optimized and some hypotheses can be weakened.
【学位授予单位】：重庆大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175.25
本文编号：2321180
[Abstract]:In this paper, we study the existence of solutions for two classes of elliptic equations with nonlocal terms. First, we study the existence of solutions for Kirchhoff type elliptic equations, including ground state solutions, multiple solutions, negative energy solutions and sign changing solutions. Secondly, we discuss the existence of solutions for Schr?dinger-Poisson equations. The specific conclusions and usage methods are as follows. This article is divided into the following six chapters: the first chapter, introduction. This paper mainly describes the physical background and academic significance of the two kinds of elliptic equations with nonlocal terms considered in this paper, as well as the present development situation at home and abroad. Finally, the main conclusions and symbols used in this paper are briefly introduced. In chapter 2, we consider the elliptic equation of Kirchhoff type with Sobolev critical exponent. In this paper, we discuss the existence of some solutions in the bounded domain of. When N4, we use the Nehari manifold and the first concentrated compactness principle of Lions to overcome the difficulty of obtaining the boundedness of (PS) sequences in the case of Q2 and Q2 (2? 4). The local compactness of (PS) sequences is obtained by combining the second set compactness principle of Lions. Finally, the existence of the ground state solution bu of the equation is obtained. We also consider the relation between the Kirchhoff type elliptic equation and the Brezis-Nirenberg problem when b0, and we obtain that the ground state solution of the Kirchhoff type elliptic equation converges to the ground state solution of the Brezis-Nirenberg problem when b0. When Q2, a truncation technique is used to truncate the energy functional, and then Krasnoselskii's genus (2? Index) theory, the equation has infinitely many negative energy solutions. The existence of positive solutions to the equation is proved when N5 satisfies some hypotheses for any Q? (1 ~ 2 *) and a b. In chapter 3, we discuss the existence of sign change solutions for Kirchhoff type elliptic equations with pure power nonlinear terms. We use the variational method and the descending flow of invariant sets and 2? Based on the index theory, it is obtained that the equation has infinitely high energy sign solutions when p _ (2 ~ (2) ~ (6). Our work extends the existing literature only when p? (4 ~ (6) has a sign solution. In chapter 4, we discuss the existence of sign variation solutions for Kirchhoff type elliptic equations with general nonlinear terms. By using a truncation technique and the Nehari manifold of sign variation and the principle of centralization compactness, it is proved that the equation has a minimum energy sign solution and a ground state solution of invariant sign. Moreover, the energy of the signed solution is strictly larger than that of the ground state solution. In chapter 5, we study another type of elliptic equations with nonlocal terms, that is, Schr?dinger-Poisson equations with Sobolev critical exponents. First, the Lax-Milgram theorem shows that the Schr?dinger-Poisson equations can be transformed into a single equation containing nonlocal terms, and then the energy functional is obtained by using the variational method. When p? 1, the energy functional is confined to the Nehari manifold, the boundedness of the (PS) sequence is obtained, and the local compactness of the (PS) sequence is obtained by using the centralization compactness principle. Finally, it is proved that the constraint of the Nehari manifold on the energy functional is a natural constraint. Therefore, the existence of a ground state solution for the original equations is obtained. When p? (0 ~ 1), a truncation technique is used to truncate the energy functional, and then 2? In the index theory, we prove that there are infinite negative energy solutions to the equation. In the sixth chapter, we summarize this paper briefly, and hope that some conclusions in this paper can be further optimized and some hypotheses can be weakened.
【学位授予单位】：重庆大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175.25
【参考文献】
相关期刊论文 前1条
1 ;Multiplicity of Solutions for a Class of Kirchhoff Type Problems[J];Acta Mathematicae Applicatae Sinica(English Series);2010年03期
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