Abstract
We study the evolution and qualitative behaviors of bifurcation curves of positive solutions for {(- u <sup>″</sup> (x) = λ f <sub>q, p</sub> (u) = λ (u <sup>q</sup> (1 - sin u) + u <sup>p</sup> ), - 1 < x < 1,; u (- 1) = u (1) = 0,) where λ > 0 is a bifurcation parameter, q < 1 is a positive bifurcation parameter, and p ≥ 1 is an evolution parameter. We prove that, for given q < 1, there exist numbers p <sup>*</sup> (q) > p <sub>*</sub> (q) > 1 such that, on the (λ, {norm of matrix} u {norm of matrix} <sub>∞</sub> )-plane, the bifurcation curve has exactly one turning point where the curve turns to the left for p > p <sup>*</sup> (q), it has at least three turning points for 1 < p < p <sub>*</sub> (q), and it has infinitely many turning points for p = 1. Hence we are able to determine the (exact) number of positive solutions. In particular we give complete descriptions of the structure of bifurcation curves when p > p <sup>*</sup> (q). Our results extend some results of Wang [S.-H. Wang, On the evolution and qualitative behaviors of bifurcation curves for a boundary value problem, Nonlinear Anal. 67 (2007) 1316-1328] from q = 1 to 0 < q ≤ 1. © 2007 Elsevier Ltd. All rights reserved.