GLOBAL BIFURCATION AND EXACT MULTIPLICITY OF POSITIVE SOLUTIONS FOR A POSITONE PROBLEM WITH CUBIC NONLINEARITY AND THEIR APPLICATIONS

We study the global bifurcation and exact multiplicity of positive solutions of { u '' (x) + lambda f(epsilon)(u) = 0, -1 < x < 1, u(-1) = u(1) = 0, f(epsilon) (u) = -epsilon u(3) + sigma u(2) + tau u + rho, where lambda, epsilon > 0 are two bifurcation parameters, and sigma, rho > 0, tau >= 0 are constants. By developing some new time-map techniques, we prove the global bifurcation of bifurcation curves for varying epsilon > 0. More precisely, we prove that, for any sigma, rho > 0, tau >= 0, there exists epsilon* > 0 such that, on the (lambda, parallel to u parallel to(infinity))- plane, the bifurcation curve is S-shaped for 0 < epsilon < epsilon* and is monotone increasing for epsilon >= epsilon*. (We also prove the global bifurcation of bifurcation curves for varying lambda > 0.) Thus we are able to determine the exact number of positive solutions by the values of epsilon and lambda. We give an application to prove a long-standing conjecture for global bifurcation of positive solutions for the problem { u '' (x) + lambda(-epsilon u(3) + u(2) + u + 1) = 0, -1 < x < 1, u(-1) = u(1) = 0, which was studied by Crandall and Rabinowitz (Arch. Rational Mech. Anal. 52 (1973), p. 177). In addition, we give an application to prove a conjecture of Smoller and Wasserman (J. Differential Equations 39 (1981), p. 283, lines 2-3) on the maximum number of positive solutions of a positone problem.