| Summary: | 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,
$$\displaylines{
u''(x)+\lambda\exp\big(\frac{au}{a+u}\big) =0,\quad 0<x< 1,\cr
u(0)=0,\quad u'(1)=-c<0,
}$$
where $4\leq a<a_1\approx4.107$. We prove that, for
$4\leq a<a_1$, there exist two nonnegative $c_0=c_0(a)<c_1=c_1(a)$
satisfying $c_0>0$ 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}.
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