Electron. J. Differential Equations, Vol. 2017 (2017), No. 61, pp. 1-12.

Classification and evolution of bifurcation curves for the one-dimensional perturbed Gelfand equation with mixed boundary conditions II

Yu-Hao Liang, Shin-Hwa Wang

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,
$$\displaylines{ u''(x)+\lambda\exp\big(\frac{au}{a+u}\big) =0,\quad 0<x<1,\cr u(0)=0,\quad u'(1)=-c<0, }$$
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 [8].

Submitted November 30, 2016. Published February 28, 2017.
Math Subject Classifications: 34B18, 74G35.
Key Words: Multiplicity; positive solution; perturbed Gelfand equation; S-shaped bifurcation curve; C-shaped bifurcation curve; time map.

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Yu-Hao Liang
Department of Applied Mathematics
National Chiao Tung University
Hsinchu 300, Taiwan
email: yhliang@nctu.edu.tw
Shin-Hwa Wang
Department of Mathematics
National Tsing Hua University
Hsinchu 300, Taiwan
email: shwang@math.nthu.edu.tw

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