摘要
We study the global bifurcation and exact multiplicity of positive solutions for { u (x) + λ f (u) = 0, − 1 < x < 1, u(−1) = u(1) = 0, where λ > 0 is a bifurcation parameter, ε (formula presented) Θ is an evolution parameter, and Θ≡ (σ , σ ) is an open interval with 0 ≤ σ < σ ≤ ∞. Under some suitable hypotheses on f , we prove that there exists ε (formula presented) Θ such that, on the (λ, (formula presented)u(formula presented) )-plane, the bifurcation curve is S-shaped for σ < ε < ε and is monotone increasing for ε ≤ ε < σ . We give an application to prove global bifurcation of bifurcation curves for the one-dimensional perturbed Gelfand problem.