We study the global bifurcation and exact multiplicity of positive solutions ofwhere 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 * > 0 such that, on the (λ, ||u|| ∞ )plane, the bifurcation curve is S-shaped for 0 < ε < ε * and is monotone increasing for ε ≥ ε * . (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 problemwhich 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.
We study the bifurcation curve and exact multiplicity of positive solutions of the positone problemwhere λ > 0 is a bifurcation parameter, f ∈ C 2 [0, ∞) satisfies f (0) > 0 and f (u) > 0 for u > 0, and f is convex-concave on (0, ∞). Under a mild condition, we prove that the bifurcation curve is S-shaped on the (λ, u ∞ )-plane. We give an application to the perturbed Gelfand problemwhere a > 0 is the activation energy parameter. We prove that, if a a * ≈ 4.166, the bifurcation curve is S-shaped on the (λ, u ∞ )-plane. Our results improve those in [S.-H. Wang, On S-shaped bifurcation curves, Nonlinear Anal. 22 (1994) 1475-1485] and [P. Korman, Y. Li, On the exactness of an S-shaped bifurcation curve,
We give a necessary and sufficient condition for W (A) to be an elliptic disk. The criterion is applied to prove the elliptic numerical ranges of certain bordered matrices.
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