Abstract
We study the evolution and qualitative behaviors of bifurcation curves of positive solutions for –u〞(x) = λ f_{q,p} (u) = λ[ u^q (1– sin u)＋u^p ], –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_(q) ＞ p_(q) ＞ 1 such that, on the ( λ,∥u∥_∞)-plane, the bifurcation curve has exactly one turning point where the curve turns to the left for p ＞ p_(q), it has at least three turning points for 1 ＜ p ＜ p_(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_(q). Our results extend some results of Wang [Nonlinear Anal. 67 (2007) 1316–1328] from q ＝ 1 to 0 ＜ q ≦ 1.