\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

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

  • * Corresponding author

    * Corresponding author
Abstract Full Text(HTML) Figure(2) Related Papers Cited by
  • We consider the bifurcation problem

    $-u''\left(t \right) = \lambda \left(u\left(t \right)+g\left(u\left(t \right) \right) \right),\;u\left(t \right)>0,\;\;t\in I: = \left(-1, 1 \right),\;\; u\left(\pm 1 \right) = 0, $

    where $g(u) ∈ C^1(\mathbb{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 $α = \Vert u_λ\Vert_∞$ 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 $α \gg 1$. 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 $\alpha \gg 1$ or not.

    Mathematics Subject Classification: Primary: 34C23, 34F10; Secondary: 34B15.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  λ(g; α) with (OP)

    Figure 2.  graph of sinx + $\epsilon\psi$(x)

  • [1] A. AmbrosettiH. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543. 
    [2] S. Cano-Casanova and J. López-Gómez, Existence, uniqueness and blow-up rate of large solutions for a canonical class of one-dimensional problems on the half-line, J. Differential Equations, 244 (2008), 3180-3203. 
    [3] S. Cano-Casanova and J. López-Gómez, Blow-up rates of radially symmetric large solutions, J. Math. Anal. Appl., 352 (2009), 166-174. 
    [4] S. ChenJ. Shi and J. Wei, Bifurcation analysis of the Gierer-Meinhardt system with a saturation in the activator production, Appl. Anal., 93 (2014), 1115-1134. 
    [5] Y. J. Cheng, On an open problem of Ambrosetti, Brezis and Cerami, Differential Integral Equations, 15 (2002), 1025-1044. 
    [6] R. Chiappinelli, On spectral asymptotics and bifurcation for elliptic operators with odd superlinear term, Nonlinear Anal., 13 (1989), 871-878. 
    [7] J. M. FraileJ. López-Gómez and J. Sabina de Lis, On the global structure of the set of positive solutions of some semilinear elliptic boundary value problems, J. Differential Equations, 123 (1995), 180-212. 
    [8] A. GalstianP. Korman and Y. Li, On the oscillations of the solution curve for a class of semilinear equations, J. Math. Anal. Appl., 321 (2006), 576-588. 
    [9] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products. Translated from the Russian. Translation edited and with a preface by Daniel Zwillinger and Victor Moll. Eighth edition. Elsevier/Academic Press, Amsterdam, 2015.
    [10] P. Korman and Y. Li, Exact multiplicity of positive solutions for concave-convex and convex-concave nonlinearities, J. Differential Equations, 257 (2014), 3730-3737. 
    [11] P. Korman and Y. Li, Computing the location and the direction of bifurcation for sign changing solutions, Differ. Equ. Appl., 2 (2010), 1-13. 
    [12] P. Korman and Y. Li, Infinitely many solutions at a resonance, Electron. J. Differ. Equ. Conf. 05, 105–111.
    [13] P. Korman, An oscillatory bifurcation from infinity, and from zero, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 335-345. 
    [14] P. Korman, Global Solution Curves for Semilinear Elliptic Equations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012.
    [15] T. Laetsch, The number of solutions of a nonlinear two point boundary value problem, Indiana Univ. Math. J., 20 (1970/1971), 1-13. 
    [16] T. Shibata, New method for computing the local behavior of $L_q$-bifurcation curve for logistic equations, Int. J. Math. Anal., 7 (2013), 29-1541. 
    [17] T. Shibata, S-shaped bifurcation curves for nonlinear two-parameter problems, Nonlinear Anal., 95 (2014), 796-808. 
    [18] T. Shibata, Asymptotic length of bifurcation curves related to inverse bifurcation problems, J. Math. Anal. Appl., 438 (2016), 629-642. 
    [19] T. Shibata, Oscillatory bifurcation for semilinear ordinary differential equations, Electron. J. Qual. Theory Differ. Equ., 44 (2016), 1-13. 
  • 加载中

Figures(2)

SHARE

Article Metrics

HTML views(429) PDF downloads(211) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return