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

Tzeng, Chih-Chun; Hung, Kuo-Chih; Wang, Shin‐Hwa

doi:10.1016/j.jde.2012.02.020

Cited by 12 publications

(14 citation statements)

References 8 publications

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“…Since B(α, u) > 0 for 0 < u < α ≤ τ , and by (18) and (19), we see that, for α ∈ (0, τ ] and λ > 0, ∂ ∂λ…”

Section: Shao-yuan Huangmentioning

confidence: 82%

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

Huang¹

2019

Discrete &Amp; Continuous Dynamical Systems - A

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In this paper, we study the classification and evolution of bifurcation curves of positive solutions for one-dimensional Minkowski-curvature problem − u / 1 − u 2 = λf (u), in (−L, L) , u(−L) = u(L) = 0, where λ > 0 is a bifurcation parameter, L > 0 is an evolution parameter, f ∈ C[0, ∞) ∩ C 2 (0, ∞) and there exists β > 0 such that (β − z) f (z) > 0 for z = β. In particular, we find that the bifurcation curve S L is monotone increasing for all L > 0 when f (u)/u is of Logistic type, and is either ⊂-shaped or S-shaped for large L > 0 when f (u)/u is of weak Allee effect type. Finally, we can apply these results to obtain the global bifurcation diagrams in some important applications including ecosystem model.

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“…Since B(α, u) > 0 for 0 < u < α ≤ τ , and by (18) and (19), we see that, for α ∈ (0, τ ] and λ > 0, ∂ ∂λ…”

Section: Shao-yuan Huangmentioning

confidence: 82%

“…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.…”

Section: Shao-yuan Huangmentioning

confidence: 99%

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

Huang¹

2019

Discrete &Amp; Continuous Dynamical Systems - A

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“…, we obtain that σ > 0 and 0 < τ ≤ 15 49 σ 2 , and hence 0 < B 1 ≤ 1 7 σ and 3 7 σ ≤ C 2 < 2 3 σ. Thus by (5), (7), (8) and θ (B 1 ) = θ (C 1 ) = 0, we obtain that…”

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confidence: 91%

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

Huang¹,

Lee²,

Yeh³

2016

CPAA

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We study exact multiplicity and bifurcation curves of positive solutions of the boundary value problemwhere σ, τ ∈ R, ρ ≥ 0, and λ > 0 is a bifurcation parameter. Then on the (λ, ||u||∞)-plane, we give a classification of four qualitatively different bifurcation curves: an S-shaped curve, a broken S-shaped curve, a ⊂-shaped curve and a monotone increasing curve.2000 Mathematics Subject Classification. Primary: 34B15, 34B18.

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“…Tzeng et al [9] also studied the global bifurcation and exact multiplicity of positive solutions of Dirichlet problem with cubic nonlinearity + (− 3 + 2 − + ) = 0, − 1 < < 1, (−1) = (1) = 0,…”

Section: Introductionmentioning

confidence: 99%

“…Equation (3) is an autonomous system, and the time-map method has been successfully employed to solve the problem (3), but it is not applicable to study the nonautonomous system (1). There are many results on exact multiplicity of solutions for the Dirichlet problems; see [9][10][11][12]. In [13], under Neumann boundary value conditions, the bifurcation of solutions to a logistic equation with harvesting has been investigated using the uniform antimaximum principle and Crandall-Rabinowitz bifurcation theorem.…”

Section: Introductionmentioning

confidence: 99%

Reversed S-Shaped Bifurcation Curve for a Neumann Problem

Xing

Chen

Yao

2018

Discrete Dynamics in Nature and Society

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We study the bifurcation and the exact multiplicity of solutions for a class of Neumann boundary value problem with indefinite weight. We prove that all the solutions obtained form a smooth reversed S-shaped curve by topological degree theory, CrandallRabinowitz bifurcation theorem, and the uniform antimaximum principle in terms of eigenvalues. Moreover, we obtain that the equation has exactly either one, two, or three solutions depending on the real parameter. The stability is obtained by the eigenvalue comparison principle.

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

Cited by 12 publications

References 8 publications

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

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

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

Reversed S-Shaped Bifurcation Curve for a Neumann Problem

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