Abstract
We study the global bifurcation and exact multiplicity of positive solutions of where λ, ε > 0 are two bifurcation parameters, and σ, ρ > 0, τ ≥ 0 are constants. By developing some new time-map techniques, we prove the global bifurcation of bifurcation curves for varying ε > 0. More precisely, we prove that, for any σ, ρ > 0, τ ≥ 0, there exists ε <sup>*</sup> > 0 such that, on the (λ, ||u||∞)- plane, the bifurcation curve is S-shaped for 0 < ε < ε <sup>*</sup> and is monotone increasing for ε ≥ ε <sup>*</sup> . (We also prove the global bifurcation of bifurcation curves for varying λ > 0.) Thus we are able to determine the exact number of positive solutions by the values of ε and λ. We give an application to prove a long-standing conjecture for global bifurcation of positive solutions for the problem 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. © 2012 American Mathematical Society.