Abstract
We study the structure of solution set of the nonlinear two-point boundary value problem {u″ (x) + f <sub>λ</sub> (u(x)) = 0, -1 < x < 1, u(-1) = u(1) = 0, where λ > 0 is a bifurcation parameter and f <sub>λ</sub> (u) = λ ∑ <sub>i=1</sub> <sup>m</sup> a <sub>i</sub> u <sup>qi</sup> + ∑ <sub>j=1</sub> <sup>n</sup> b <sub>j</sub> u <sup>pj</sup> 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.