【作者】 杨小飞；

【导师】 徐本龙；

【作者基本信息】 上海师范大学 ， 基础数学， 2009， 硕士

【摘要】 本文研究的方程形如:u″(x) +λf(u(x)) = 0, -1≤x≤1;u(-1) = u(1) = 0.其中函数f(u(x))的形式在不同问题中不同,例如在研究气体燃烧的稳态状态时,对应函数,其中a是参数;在研究反应扩散方程时,对应函数f(u(x)) = u - au - bc - u ,其中a,b,c都是参数;在很多几何和数学物理分支问题中,对应函数f(u(x)) = u~p + u~q ,其中p,q都是参数, 0 < p < 1 < q.等等.对该类方程解的研究是微分方程学科的重要分支,经分析知方程的解和参数λ取值及函数f(u(x))的性质都有很大关系,本文重点研究的就是当函数f(u)是凹凸类型时对应的方程解的确切个数.经过阅读大量的相关文献([1] [6],[13],[20],[21],[38] [41],[43],[46],[47]),我们知道方程的确切解个数的确定是个很不容易的问题([8],[15],[17],[24],[28],[37],[43]),目前只有对特殊的函数f(u(x))的一些结论.对上述问题的研究目前采用的方法主要有:Time ? map方法和分歧方法两种.前一种Time ? map方法是一种长久以来普遍使用的方法,例如,在TheodoreLaetsch,J.SmollerandA.Wasserman,S. ?H,Wang ,等人的文章中普遍出现(参见文献([8][15][13][17])等).分歧方法是由Y iLi,TianchengOuyang,JunPingShi,PhilipKorman等发展起来的,它适用于研究Rn内球上的半线性椭圆方程([1][2][5][6][10][24][38][39][40][43]).本论文用Time ? map的方法研究一类半线性微分方程的确切解个数问题.解决了当非线性项为两种不同情形下的微分方程的确切解个数问题,同时解决了用Time ? map方法确定解曲线的临界点处转向的问题.在高维情形中,证明线性化方程的解为正解是问题的最难点([3][4][38]).在文献[3]中,作者采用一个重要的恒等式较容易地解决了这个问题,并采用Time ? map方法得到确切解的结论.在文献[4]中作者利用极值原理和一系列巧妙的构造证明了线性化方程的解为正解,并利用变分方法得到了关于方程的确切解个数的一些结论,同时也提出了一些开放式问题,其中之一是对该问题相应一维情形下的解曲线的描述,本论文第三章命题二正是对该问题的较好的解决,本论文其它部分的内容安排:第一章,绪论;第二章,含有超线性凹凸项的半线性微分方程解的结构;第三章,含有超线性次线性混合项的半线性微分方程解的结构.更多还原

【Abstract】 In this paper, I am going to discuss the equation :u″(x) +λf(u(x)) = 0,-1≤1;u(-1) = u(1) = 0.in which the function f(u) has different forms in different questions. For example , when we dis-cuss Gaseous combustion’s Stable state condition, the function . When we discussreactive diffusion equation,the function f(u(x)) = u - au - bu - c; In geometry or mathematicsphysics domain, the function f(u(x)) = u~p + u~q, 0 < p < 1 < q. And so on. It is one of the mostimportant branches, that many people are devoted in researching this kind of equations’solutions.After careful analysis , we can find the Solution integer is close related with parameterλand thefunction f(u). In this paper, I am going to discuss the exact multiplicity when the equation is likeconcave-convex. After reading massive references ([1] [6], [13], [20], [21], [38] [41], [43], [46], [47]),we know this is not easy([8], [15], [17], [24], [28], [37], [43]). To present , there is few conclusionin view of some certain of equations. The most two important methods is Time-map methodand bifurcation method. The Preceding kind is used for many years. For example , it appears inmany papers from Theodore Laetsch, J.Smoller and A.Wasserman, S.-H,Wang([8][15][13][17])).Bifurcation method is developed by YiLi, Tiancheng Ouyang, JunPing Shi, Philip Korman. Thismethod is mostly used searching for semi-linear elliptic differential equations in a n-dimensionalball([1][2][5][6][10][24][38][39][40][43])). In this paper, I am going to use Time-map method tosolve the exact multiplicities of a kind of semi-linear elliptic differential equations. I solve thecorresponding problem containing two kinds of different f(x). What’s more, I solve the problemhow to use the Time-map method to get the change direction on a critical point. When the dimen-sion is three or more , proving the linear equations’solutions is positive is the most difficulty . Inthe paper [3],the author easily solved this question with the aid of an identical equation . In thepaper [4], the author stated some open questions after obtaining a good conclusion. I am goingto discuss the solutions’curve in one dimension. In Chapter 1, Introduction ; In Chapter 2, thesolution’s structure of semi-linear differential equations containing super linear items; In Chap-ter 3, the solution’s structure of semi-linear differential equations containing linear-super linearconcave-convex functions.更多还原