Taiwanese Journal of Mathematics

Classification and Evolution of Bifurcation Curves for a Dirichlet-Neumann Boundary Value Problem and its Application

Da-Chang Kuo, Shin-Hwa Wang, and Yu-Hao Liang

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Abstract

We study the classification and evolution of bifurcation curves of positive solutions for the one-dimensional Dirichlet-Neumann boundary value problem \[ \begin{cases} u''(x) + \lambda f(u) = 0, \quad 0 \lt x \lt 1, \\ u(0) = 0, \quad u'(1) = -c \lt 0, \end{cases} \] where $\lambda \gt 0$ is a bifurcation parameter and $c \gt 0$ is an evolution parameter. We mainly prove that, under some suitable assumptions on $f$, there exists $c_{1} \gt 0$, such that, on the $(\lambda,\|u\|_{\infty})$-plane, (i) when $0 \lt c \lt c_{1}$, the bifurcation curve is $S$-shaped; (ii) when $c \geq c_{1}$, the bifurcation curve is $\subset$-shaped. Our results can be applied to the one-dimensional perturbed Gelfand equation with $f(u) = \exp \big( \frac{au}{a+u} \big)$ for $a \geq 4.37$.

Article information

Source
Taiwanese J. Math., Volume 23, Number 2 (2019), 307-331.

Dates
Received: 7 October 2017
Revised: 6 April 2018
Accepted: 10 April 2018
First available in Project Euclid: 24 May 2018

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1527127365

Digital Object Identifier
doi:10.11650/tjm/180502

Mathematical Reviews number (MathSciNet)
MR3936002

Subjects
Primary: 34B18: Positive solutions of nonlinear boundary value problems 74G35: Multiplicity of solutions

Keywords
bifurcation multiplicity positive solution $S$-shaped bifurcation curve $\subset$-shaped bifurcation curve time map

Citation

Kuo, Da-Chang; Wang, Shin-Hwa; Liang, Yu-Hao. Classification and Evolution of Bifurcation Curves for a Dirichlet-Neumann Boundary Value Problem and its Application. Taiwanese J. Math. 23 (2019), no. 2, 307--331. doi:10.11650/tjm/180502. https://projecteuclid.org/euclid.twjm/1527127365


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References

  • K. J. Brown, M. M. A. Ibrahim and R. Shivaji, $S$-shaped bifurcation curves, Nonlinear Anal. 5 (1981), no. 5, 475–486.
  • A. K. Drame and D. G. Costa, On positive solutions of one-dimensional semipositone equations with nonlinear boundary conditions, Appl. Math. Lett. 25 (2012), no. 12, 2411–2416.
  • J. Goddard II, E. K. Lee and R. Shivaji, A double $S$-shaped bifurcation curve for a reaction-diffusion model with nonlinear boundary conditions, Bound. Value Probl. 2010 (2010), Art. ID 357542, 23 pp.
  • P. V. Gordon, E. Ko and R. Shivaji, Multiplicity and uniqueness of positive solutions for elliptic equations with nonlinear boundary conditions arising in a theory of thermal explosion, Nonlinear Anal. Real World Appl. 15 (2014), 51–57.
  • S.-Y. Huang and S.-H. Wang, Proof of a conjecture for the one-dimensional perturbed Gelfand problem from combustion theory, Arch. Ration. Mech. Anal. 222 (2016), no. 2, 769–825.
  • K.-C. Hung and S.-H. Wang, A theorem on $S$-shaped bifurcation curve for a positone problem with convex-concave nonlinearity and its applications to the perturbed Gelfand problem, J. Differential Equations 251 (2011), no. 2, 223–237.
  • K.-C. Hung, S.-H. Wang and C.-H. Yu, Existence of a double $S$-shaped bifurcation curve with six positive solutions for a combustion problem, J. Math. Anal. Appl. 392 (2012), no. 1, 40–54.
  • P. Korman and Y. Li, On the exactness of an $S$-shaped bifurcation curve, Proc. Amer. Math. Soc. 127 (1999), no. 4, 1011–1020.
  • D.-C. Kuo, S.-H. Wang and Y.-H. Liang, Proofs of some results in the article: Classification and evolution of bifurcation curves for a Dirichlet-Neumann boundary value problem and its application, Available from: http://www.math.nthu.edu.tw/\symbol126shwang/PfEvolBifDiriNeuMixBC.pdf.
  • T. Laetsch, The number of solutions of a nonlinear two point boundary value problem, Indiana Univ. Math. J. 20 (1970), 1–13.
  • Y.-H. Liang and S.-H. Wang, Classification and evolution of bifurcation curves for the one-dimensional perturbed Gelfand equation with mixed boundary conditions, J. Differential Equations 260 (2016), no. 12, 8358–8387.