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September  2018, 17(5): 2139-2147. doi: 10.3934/cpaa.2018102

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

Tetsutaro Shibata ,

Laboratory of Mathematics, Graduate School of Engineering, Hiroshima University, Higashi-Hiroshima, 739-8527, Japan

* Corresponding author

Received  February 2017 Revised  June 2017 Published  April 2018

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.
Keywords: global and local structure of bifurcation curves, periodic nonlinear terms.
Mathematics Subject Classification: Primary: 34C23, 34F10; Secondary: 34B15.
Citation: Tetsutaro Shibata. Global behavior of bifurcation curves for the nonlinear eigenvalue problems with periodic nonlinear terms. Communications on Pure and Applied Analysis, 2018, 17 (5) : 2139-2147. doi: 10.3934/cpaa.2018102
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. 

Figure 1.  λ(g; α) with (OP)
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Figure 2.  graph of sinx + $\epsilon\psi$(x)
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