我們研究一維半正狄利克雷邊界問題的正解數與分枝曲線的演化與結構。
u''(x)+λg(u)-μ=0,u(-1)=u(1)=0,
這裡的λ,μ>0 為兩個分枝參數。假設非線性項g滿足g(0)=g(1)=0≥0, g(u)>0在(0,1)且g是凹函數在(0,1)或(是凹-凸函數在(0,1)且滿足一些特定條件。我們證明在(λ,‖u‖∞)平面上，對於固定μ>0, 分枝曲線包含一條⊂型曲線和其後我們研究對於改變μ>0,分枝曲線的結構與演化。(我們亦證明在(μ,‖u‖∞)平面上， 對於固定λ>π²/(4g'(0)),分枝曲線包含一條反向⊂型曲線且其後我們研究對於改變λ>π²/(4g'(0)) 分枝曲線的結構與演化。)我們亦研究在(μ,λ,‖u‖∞)空間上分枝曲面的結構與形狀。

We study the structures and evolution of bifurcation diagrams and exact multiplicity of positive solutions for the one-dimensional semipositone Dirichlet problem
u''(x)+λg(u)-μ=0,u(-1)=u(1)=0,
where λ,μ>0 are two bifurcation parameter. We assume that nonlinearity g satisfies g(0)=g(1)=0≥0, g(u)>0 on (0,1) and g either is concave on (0,1) or (is concave-convex on (0,1) and satisfies a certain condition). We prove that, for any fixed μ>0, the bifurcation diagram always consists of a ⊂-shaped curve on the (λ,‖u‖_∞)-plane and then we study the structures and evolution of bifurcation diagrams for varying μ>0. (We also prove that, for any fixed λ>π²/(4g'(0)), the bifurcation diagram always consists of a reversed ⊂-shaped curve on the (μ,‖u‖∞)-plane and then we study the structures and evolution of bifurcation diagrams for varying λ>π²/(4g'(0)).) We also study, in the (μ,λ,‖u‖∞)-space, the shape and structure of the bifurcation surface.

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