一類三次非線性正定問題全分枝性及確切正解個數

Abstract

本篇論文主要是探討一類三次非線性正定問題的全分支性及正解的確切個數。在 適當的條件下，我們利用時間映射(time map)的方法來研究此一問題，並且證明在不同的演化參數下會有不同的分支曲線圖，進一步來說這些分支曲線基本上有兩種，不是單調曲線就是我們所稱的 S 型曲線

We study the global bifurcation and exact multiplicity of positive solutions of

Where λ,ε>0 are two bifurcation parameters, and σ,ρ>0, 0<κ √σρ are constants. We prove the global bifurcation of bifurcation curves for varying ε>0 by developed some time-map techniques. More precisely, we prove that, for anyσ,ρ>0, 0<κ √σρ, there exists ε ̃>0 such that, on the (λ,‖u‖_∞ )-plane, the bifurcation curve is S-shaped for 0<ε<ε ̃ and is monotone increasing forε ε ̃. (We also prove the global bifurcation of bifurcation curves for varyingλ>0.) Thus we are able to determine the exact number of positive solutions by the values of ε andλ. Our results extend those of Hung and Wang ( Trans. Amer. Math. Soc., accepted to appear under minor revision ) from κ 0 to 0<κ √σρ.