S-shaped bifurcation curves for nonlinear two-parameter problems

Shibata, Tetsutaro

doi:10.1016/j.na.2013.10.015

Cited by 4 publications

(2 citation statements)

References 12 publications

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“…The method to study the local behavior of λ(α) has been already established in [17] and [18], since the time-map method and Taylor expansion work very well in this case. To understand the total structure of λ(g, α), we show the following asymptotic formulas for completeness.…”

Section: Corollary 12 Assume That G(u) Satisfies (A1)-(a2)mentioning

confidence: 99%

Global behavior of bifurcation curves for the nonlinear eigenvalue problems with periodic nonlinear terms

Shibata¹

2018

Communications on Pure &Amp; Applied Analysis

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We consider the bifurcation problem −u ′′ (t) = λ (u(t) + g(u(t))) , u(t) > 0, t ∈ I := (−1, 1), u(±1) = 0, where g(u) ∈ C 1 (R) is a periodic function with period 2π and λ > 0 is a bifurcation parameter. It is known that, under the appropriate conditions on g, λ is parameterized by the maximum norm α = ∥u λ ∥∞ of the solution u λ associated with λ and is written as λ = λ(α). If g(u) is periodic, then it is natural to expect that λ(α) is also oscillatory for α ≫ 1. We give a simple condition of g(u), by which we can easily check whether λ(α) is oscillatory and intersects the line λ = π 2 /4 infinitely many times for α ≫ 1 or not.

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Section: Corollary 12 Assume That G(u) Satisfies (A1)-(a2)mentioning

confidence: 99%

Global behavior of bifurcation curves for the nonlinear eigenvalue problems with periodic nonlinear terms

Shibata¹

2018

Communications on Pure &Amp; Applied Analysis

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“…For other results concerning the existence of an S-shaped connected component in the set of solutions for diverse boundary value problems, see [23][24][25][26] for the semilinear boundary value problems, and [27] for the p-Laplacian boundary value problems.…”

Section: Introductionmentioning

confidence: 99%

S-shaped connected component of radial positive solutions for a prescribed mean curvature problem in an annular domain

2019

Open Mathematics

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In this paper, we show the existence of an S-shaped connected component in the set of radial positive solutions of boundary value problem$$\begin{array}{} \displaystyle \left\{\,\begin{array}{} -\text{ div}\big(\phi_N(\nabla y)\big)=\lambda a(|x|)f(y)\, \, \, \, \, \text{in}\, \, \mathcal{A},\\\frac{\partial y}{\partial \nu}=0\, \, \, \,\, \text{ on }\, \, {\it\Gamma}_1,\qquad y=0\, \, \, \, \text{ on}\, \, {\it\Gamma}_2,\\ \end{array} \right. \end{array} $$where R2 ∈ (0, ∞) and R1 ∈ (0, R2) is a given constant, 𝓐 = {x ∈ ℝN : R1 < ∣x∣ < R2}, Γ1 = {x ∈ ℝN : ∣x∣ = R1}, Γ2 = {x ∈ ℝN : ∣x∣ = R2}, $\begin{array}{} \phi_N(s)=\frac{s}{\sqrt{1-|s|^2}}, \end{array} $s ∈ ℝN, λ is a positive parameter, a ∈ C[R1, R2], f ∈ C[0, ∞), $\begin{array}{} \frac{\partial y}{\partial \nu} \end{array} $ denotes the outward normal derivative of y and ∣⋅∣ denotes the Euclidean norm in ℝN. The proof of main result is based upon bifurcation techniques.

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A complete classification of bifurcation diagrams for a class of (p,q)-Laplace equations

Kajikiya

Sim

Tanaka

2018

Journal of Mathematical Analysis and Applications

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S-shaped bifurcation curves for nonlinear two-parameter problems

Cited by 4 publications

References 12 publications

Global behavior of bifurcation curves for the nonlinear eigenvalue problems with periodic nonlinear terms

Global behavior of bifurcation curves for the nonlinear eigenvalue problems with periodic nonlinear terms

S-shaped connected component of radial positive solutions for a prescribed mean curvature problem in an annular domain

A complete classification of bifurcation diagrams for a class of (p,q)-Laplace equations

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