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

Abstract

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.