Global bifurcation and exact multiplicity of positive solutions for a positone problem with cubic nonlinearity and their applications

Abstract

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.

Bifurcation diagrams of positive solutions for one-dimensional Minkowski-curvature problem and its applications

“…from which it follows that there exists t * ∈ (0, 1) such that B(β, βt) < 1 for t * < t < 1. By (11) and (16), we see that…”

“…There are many references in studying the bifurcation curveS of (3), cf. [2,10,11,18,12,15,16,21,22]. For instance, Ouyang and Shi [15] obtained the bifurcation diagrams for the problem (3) with nonlinearity f (u) = u p − q q , 1 < p < q.…”

“…Clearly, there exists β > 0 such that f (u) > 0 on (0, β), f (β) = 0 and f (u) < 0 on (β, ∞), cf. [11,Theorem 2.1]. We compute that η = ∞.…”

“…It implies that (C2) holds. By [11,Theorem 2.1], there exists a 0 > 0 such thatS is S-shaped for 0 < a < a 0 and monotone increasing for a ≥ a 0 . Then we assert that (H3) holds.…”

Classification of bifurcation curves of positive solutions for a nonpositone problem with a quartic polynomial

“…They studied exact multiplicity and bifurcation curves of positive solutions of the problem (1) with cubic polynomials f (u) = −(u − a)(u − b)(u − c), a < b < c (Note that f changes sign at most once on (0, ∞).). In addition, Wang [9] gave a correction for [7], and Hung and Wang [3] extended partial results of [7]. Smoller and Wasserman [7], Wang [9], and Hung and Wang [3] obtained that such that, on the (λ, u ∞ )-plane, (i) for 0 < ε <ε, the bifurcation curveS of (1) is an S-shaped curve, (ii) for ε ≥ε, the bifurcation curveS of (1) is a monotone increasing curve.…”