Taiwanese Journal of Mathematics

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

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
Revised: 6 April 2018
Accepted: 10 April 2018
First available in Project Euclid: 24 May 2018

https://projecteuclid.org/euclid.twjm/1527127365

Digital Object Identifier
doi:10.11650/tjm/180502

Mathematical Reviews number (MathSciNet)
MR3936002

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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