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April, 2019 Classification and Evolution of Bifurcation Curves for a Dirichlet-Neumann Boundary Value Problem and its Application
Da-Chang Kuo, Shin-Hwa Wang, Yu-Hao Liang
Taiwanese J. Math. 23(2): 307-331 (April, 2019). DOI: 10.11650/tjm/180502

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 $a \geq 4.37$.

Citation

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Da-Chang Kuo. Shin-Hwa Wang. Yu-Hao Liang. "Classification and Evolution of Bifurcation Curves for a Dirichlet-Neumann Boundary Value Problem and its Application." Taiwanese J. Math. 23 (2) 307 - 331, April, 2019. https://doi.org/10.11650/tjm/180502

Information

Received: 7 October 2017; Revised: 6 April 2018; Accepted: 10 April 2018; Published: April, 2019
First available in Project Euclid: 24 May 2018

zbMATH: 07055571
MathSciNet: MR3936002
Digital Object Identifier: 10.11650/tjm/180502

Subjects:
Primary: 34B18 , 74G35

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

Rights: Copyright © 2019 The Mathematical Society of the Republic of China

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Vol.23 • No. 2 • April, 2019
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