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A note concerning a property of symplectic matrices
Global behavior of bifurcation curves for the nonlinear eigenvalue problems with periodic nonlinear terms
Laboratory of Mathematics, Graduate School of Engineering, Hiroshima University, Higashi-Hiroshima, 739-8527, Japan |
$-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, $ |
$g(u) ∈ C^1(\mathbb{R}) $ |
$λ > 0$ |
$α = \Vert u_λ\Vert_∞$ |
$u_λ$ |
$λ$ |
$λ = λ(α)$ |
$g(u)$ |
$λ(α)$ |
$α \gg 1$ |
$g(u)$ |
$λ(α)$ |
$λ = π^2/4$ |
$\alpha \gg 1$ |
References:
[1] |
A. Ambrosetti, H. 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. Chen, J. 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. Fraile, J. 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. Galstian, P. 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.
|
show all references
References:
[1] |
A. Ambrosetti, H. 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. Chen, J. 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. Fraile, J. 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. Galstian, P. 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.
|


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