Taiwanese Journal of Mathematics

An Evolutionary Property of the Bifurcation Curves for a Positone Problem with Cubic Nonlinearity

Shao-Yuan Huang and Shin-Hwa Wang

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We study an evolutionary property of the bifurcation curves for a positone problem with cubic nonlinearity\[\begin{cases}  u''(x) + \lambda f(u) = 0, \quad -1 \lt x \lt 1, \\  u(-1) = u(1) = 0, \\  f(u) = -\varepsilon u^{3} + \sigma u^{2} + \tau u + \rho,\end{cases}\]where $\lambda \gt 0$ is a bifurcation parameters, $\varepsilon \gt 0$ is an evolution parameter, and $\sigma$, $\rho \gt 0$, $\tau \geq 0$ are constants. In addition, we improve lower and upper bounds of the critical bifurcation value $\widetilde{\varepsilon}$ of the problem.

Article information

Taiwanese J. Math., Volume 20, Number 3 (2016), 639-661.

First available in Project Euclid: 1 July 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

positive solution exact multiplicity turning point s-shaped bifurcation curve


Huang, Shao-Yuan; Wang, Shin-Hwa. An Evolutionary Property of the Bifurcation Curves for a Positone Problem with Cubic Nonlinearity. Taiwanese J. Math. 20 (2016), no. 3, 639--661. doi:10.11650/tjm.20.2016.6563. https://projecteuclid.org/euclid.twjm/1498874472

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