基尔霍夫—薛定谔—泊松型方程的正解与变号解

发布时间：2018-04-29 02:29

  本文选题：基尔霍夫-薛定谔-泊松型方程 + 正解　； 参考：《山东师范大学》2017年硕士论文

【摘要】：随着数学研究的不断发展,人们发现在解决物理问题时应用变分方法来研究微分方程比较方便,变分方法因此日益受到重视.变分方法的发展大致经历了两个阶段,二十世纪五十年代以前是以古典变分法为主的第一阶段,七十年代以后进入以有限元法为主的第二阶段,并从结构力学和固体力学发展到流体力学和其他领域.在进入第二阶段即有限元法为主的过程中,人们发明了山路引理和喷泉定理等临界点理论,并开始用这些临界点理论研究非线性方程问题,特别是非线性椭圆边值问题.到目前为止,在相应的方程中得出了很多有关解的有意义的结果.Kirchhoff在研究弹性带的自由振动时,第一次提出了基尔霍夫型微分方程.人们在量子力学中研究带电波与自身静电场的相互作用时,作为其物理方程第一次提出了薛定谔-泊松型方程.随着以上两种方程的研究,人们开始关注基尔霍夫-薛定谔-泊松型方程,并得到了关于解的多样性等结果.本文主要利用变分方法,山路引理,对称山路引理的变形等临界点理论,得到两类基尔霍夫-薛定谔-泊松型方程的正解,负解,变号解及无穷多变号解的存在性的结果.主要包括以下三章:第一章主要介绍了基尔霍夫-薛定谔-泊松型方程的研究现状和一些本文中常用符号及基础知识.第二章讨论了在R3上的一类基尔霍夫-薛定谔-泊松型方程:其中a0,b≥ 0,μ0,1q2,4p6.F(λ,x,u)=λf(x)|u|q-2u+g(x)|u|p-2u.利用变分方法,在某些适当的条件下,我们可以得到该问题的一个正解,它是相应地能量泛函的一个局部极小值点.利用山路引理,我们可以得到该问题的另一个不同的正解.第三章讨论了在R3上的一类基尔霍夫-薛定谔-泊松型方程:其中a＞0,b≥0, f(x,u)是R3 x R→R上面的非平凡的函数.利用山路引理,我们可以得到该问题正负解的存在性.利用变形的山路引理,我们可以得到变号解的存在性.利用对称山路引理,在一些适当的条件下,我们可以得到该问题无穷多变号解的存在性.
[Abstract]:With the development of mathematical research, it is found that it is more convenient to study differential equations by using variational method in solving physical problems. Therefore, variational methods are paid more and more attention. The development of variational methods has gone through two stages. Before the 1950s, it was the first stage dominated by classical variational methods. After the 1970s, it entered the second stage, which was dominated by finite element method. And from structural mechanics and solid mechanics to hydrodynamics and other fields. In the process of the second stage, i.e. the finite element method, people invented the critical point theory, such as mountain pass Lemma and fountain theorem, and began to use these critical point theories to study the nonlinear equations, especially the nonlinear elliptic boundary value problem. Up to now, in the corresponding equations, many meaningful results about the solutions have been obtained. Kirchhoff, in studying the free vibration of elastic bands, has put forward the Kirchhoff differential equation for the first time. The Schrodinger Poisson type equation is first proposed as the physical equation of the interaction between the band electric wave and its own electrostatic field in quantum mechanics. With the study of the above two equations, people begin to pay attention to the Kirchhoff Schrodinger Poisson type equation, and obtain the results on the diversity of solutions. In this paper, by using the critical point theory of variational method, mountain pass Lemma and symmetric mountain pass Lemma, we obtain the existence of positive solutions, negative solutions, variable sign solutions and infinite variable sign solutions for two classes of Kirchhoff Schrodinger Poisson type equations. There are three chapters as follows: the first chapter mainly introduces the research status of Kirchhoff-Schrodinger-Poisson type equation and some commonly used symbols and basic knowledge in this paper. In chapter 2, we discuss a class of Kirchhoff Schrodinger Poisson type equations on R3, where a 0b 鈮，

本文编号：1818000