發(fā)布時間：2018-04-29 02:29

  本文選題：基爾霍夫-薛定諤-泊松型方程 + 正解��； 參考：《山東師范大學(xué)》2017年碩士論文

【摘要】：隨著數(shù)學(xué)研究的不斷發(fā)展,人們發(fā)現(xiàn)在解決物理問題時應(yīng)用變分方法來研究微分方程比較方便,變分方法因此日益受到重視.變分方法的發(fā)展大致經(jīng)歷了兩個階段,二十世紀(jì)五十年代以前是以古典變分法為主的第一階段,七十年代以后進(jìn)入以有限元法為主的第二階段,并從結(jié)構(gòu)力學(xué)和固體力學(xué)發(fā)展到流體力學(xué)和其他領(lǐng)域.在進(jìn)入第二階段即有限元法為主的過程中,人們發(fā)明了山路引理和噴泉定理等臨界點(diǎn)理論,并開始用這些臨界點(diǎn)理論研究非線性方程問題,特別是非線性橢圓邊值問題.到目前為止,在相應(yīng)的方程中得出了很多有關(guān)解的有意義的結(jié)果.Kirchhoff在研究彈性帶的自由振動時,第一次提出了基爾霍夫型微分方程.人們在量子力學(xué)中研究帶電波與自身靜電場的相互作用時,作為其物理方程第一次提出了薛定諤-泊松型方程.隨著以上兩種方程的研究,人們開始關(guān)注基爾霍夫-薛定諤-泊松型方程,并得到了關(guān)于解的多樣性等結(jié)果.本文主要利用變分方法,山路引理,對稱山路引理的變形等臨界點(diǎn)理論,得到兩類基爾霍夫-薛定諤-泊松型方程的正解,負(fù)解,變號解及無窮多變號解的存在性的結(jié)果.主要包括以下三章:第一章主要介紹了基爾霍夫-薛定諤-泊松型方程的研究現(xiàn)狀和一些本文中常用符號及基礎(chǔ)知識.第二章討論了在R3上的一類基爾霍夫-薛定諤-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.利用變分方法,在某些適當(dāng)?shù)臈l件下,我們可以得到該問題的一個正解,它是相應(yīng)地能量泛函的一個局部極小值點(diǎn).利用山路引理,我們可以得到該問題的另一個不同的正解.第三章討論了在R3上的一類基爾霍夫-薛定諤-泊松型方程:其中a＞0,b≥0, f(x,u)是R3 x R→R上面的非平凡的函數(shù).利用山路引理,我們可以得到該問題正負(fù)解的存在性.利用變形的山路引理,我們可以得到變號解的存在性.利用對稱山路引理,在一些適當(dāng)?shù)臈l件下,我們可以得到該問題無窮多變號解的存在性.
[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，

本文編號：1818000