The structure of the solution set of a generalized Ambrosetti–Brezis–Cerami problem in one space variable

We study the structure of solution set of the nonlinear two-point boundary value problem u (x)+fλ(u(x)) = 0, −1 0 is a bifurcation parameter and fλ(u) = λ m i=1 aiuqi + n j=1 bj upj satisfies (A1)–(A4). Under (A1)–(A4), we prove that there exists λ∗ > 0 such that the problem has exactly two positive solutions for 0<λ<λ∗, exactly one positive solution for λ = λ∗, and no positive solution for λ > λ∗. More precisely, we give a complete description of the structure of the solution set.  2005 Elsevier Inc. All rights reserved.