Abstract
We study the evolution and qualitative behaviors of bifurcation curves of positive solutions for {(- u <sup>″</sup> (x) = λ (u (1 - sin u) + u <sup>p</sup> ), - 1 < x < 1,; u (- 1) = u (1) = 0,) where λ > 0 is a bifurcation parameter and p ≥ 1 is an evolution parameter. On the (λ, {norm of matrix} u {norm of matrix} <sub>∞</sub> )-plane, we prove that the bifurcation curve has exactly one turning point where the curve turns to the left for p > 2, it is a monotone curve for p = 2, it has at least two turning points for 1 < p ≤ over(p, ̃) ≡ frac(1 + 2 cos 1 - sin 1, 1 - 2 cos 1 + sin 1) ≈ 1.629, 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 ≥ 2. We also give some numerical simulations of bifurcation curves for p ≥ 1. © 2006 Elsevier Ltd. All rights reserved.