Abstract
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. 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