我們研究在Dirichlet-Neumann邊界條件下的正解分枝曲線的分類與演化，u''(x)+λf(u)=0, 0<x<1, u(0)=0, u'(1)=-c<0,這裡的λ>0，是分枝參數；而c>0，是演化參數。我們主要要證明函數f在適當的假設下，我們可以找到一個c₁>0，使得在(λ,‖u‖∞)平面上，我們有以下兩個性質。
(1)當0<c<c₁，分枝曲線為S型，而且在某些區間λ，會存在至少三個正解。
(2)當c≥c₁，分枝曲線為C型，而且在某些區間λ，會存在至少兩個正解。
我們的研究結果可以應用在一維的perturbed Gelfand equation，函數f(u)=exp((au)/(a+u))在a≥4.37。

We study the classification and evolution of bifurcation curves of positive solutions for the Dirichlet-Neumann boundary value problem u''(x)+λf(u)=0, 0<x<1, u(0)=0, u'(1)=-c<0, where λ>0 is a bifurcation parameter and c>0 is an evolution parameter. We mainly prove that, under some suitable assumptions on f, there exists c₁>0, such that, on the (λ,‖u‖∞)-plane, (i) when 0<c<c₁, the bifurcation curve is S-shaped and the problem has at least three positive solutions for some range of positive λ; (ii) when c≥c₁, the bifurcation curve is ⊂-shaped and the problem has at least two positive solutions for some range of positive λ. Our results can be applied to the one-dimensional perturbed Gelfand equation with f(u)=exp((au)/(a+u)) for a≥4.37.

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