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 $a \geq 4.37$.