臺灣博碩士論文加值系統

我們首先研究以下邊界爆炸值問題解存在之充分與必要條件
- u'' =λf(u(x)), 0 < x < 1,
lim u(x) = ∞ as x→0 plus
lim u(x) = ∞ as x→1 minus ,
這裡的λ是一個正的分枝參數.
再來討論 f 是局部Lipschitz連續,除了可能在某一點s滿足f(s) = 0之外, 不失一般性我們取這點s為0來討論(即f(0) = 0), 在 (ρ,λ) - 平面圖形上分析它的分枝曲線的漸近行為, 其中ρ= min u(x), x 屬於 (0,1). 因此我們可以根據λ來判定其邊界爆炸解之存在與否和對映的解之個數.

We investigate the necessary and sufficient conditions for the existence of solutions of the boundary blow-up problem
- u'' =λf(u(x)), 0 < x < 1,
lim u(x) = ∞ as x→0 plus
lim u(x) = ∞ as x→1 minus,
where λ is a positive bifurcation parameter and f is locally Lipschitz continuous at all points in R except possibly at point s = 0 and f is continuous there. We also study asymptotic behaviors of the bifurcation curve on the (ρ,λ) -plane, where
ρ= min u(x), x belong (0,1). Hence we are able to determine the number of solutions for anyλ > 0. Some interesting examples are given.

1 Introduction-------------------------------------------------2
2 Main Results------------------------------------------------5
2.1 Necessary and Sufficient Conditions----------------7
2.2 Asymptotic Behaviors of G(ρ)--------------------12
3 Lemmas----------------------------------------------------13
4 Proofs of Main Results-----------------------------------24
References----------------------------------------------------38

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