Electronic Journal of Differential Equations (Feb 2017)
Classification and evolution of bifurcation curves for the one-dimensional perturbed Gelfand equation with mixed boundary conditions II
Abstract
In this article, we study the classification and evolution of bifurcation curves of positive solutions for the one-dimensional perturbed Gelfand equation with mixed boundary conditions, $$\displaylines{ u''(x)+\lambda\exp\big(\frac{au}{a+u}\big) =0,\quad 00$ for $4\leq a<a^{\ast}\approx4.069$, and $c_0=0$ for $a^{\ast}\leq a<a_1$, such that, on the $(\lambda,\|u\|_{\infty})$-plane, (i) when $0<c<c_0$, the bifurcation curve is strictly increasing; (ii) when $c=c_0$, the bifurcation curve is monotone increasing; (iii) when $c_0<c<c_1$, the bifurcation curve is S-shaped; (iv) when $c\geq c_1$, the bifurcation curve is C-shaped. This work is a continuation of the work by Liang and Wang [8] where authors studied this problem for $a\geq a_1$, and our results partially prove a conjecture on this problem for $4\leq a<a_1$ in \cite{Liang-Wang}.
