On the evolution and qualitative behaviors of bifurcation curves for a boundary value problem

We study the evolution and qualitative behaviors of bifurcation curves of positive solutions for {-u ''(x) =lambda(u (1 - sin u) + u(p)) -1 < x < 1, u(-1) = u(1) = 0, where lambda > 0 is a bifurcation parameter and p >= 1 is an evolution parameter. On the (lambda, parallel to u parallel to(infinity))-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 <= (p) over tilde = 1+2 cos 1-sin 1/1-2 cos 1 + sin 1 approximate to 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. (c) 2006 Elsevier Ltd. All rights reserved.