Exact Multiplicity Of Semi-linear Differential Equations' Solutions Involving Concave-convex Functions

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.