Classification and evolution of bifurcation curves for the one-dimensional perturbed Gelfand equation with mixed boundary conditions II
Date
2017-02-28
Authors
Liang, Yu-Hao
Wang, Shin-Hwa
Journal Title
Journal ISSN
Volume Title
Publisher
Texas State University, Department of Mathematics
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,
u″(x) + λ exp (αu/α+u) = 0, 0 < x < 1,
u(0) = 0, u′(1) = -c < 0,
where 4 ≤ α < α1 ≈ 4.107. We prove that, for 4 ≤ α < α1, there exist two nonnegative c0 = c0(α) < c1 = c1(α) satisfying c0 > 0 for 4 ≤ α < α* ≈ 4.69, and c0 = 0 for α* ≤ α < α1, such that, on the (λ, ‖u‖∞)-plane, (i) when 0 < c < c0, the bifurcation curve is strictly increasing; (ii) when c = c0, the bifurcation curve is monotone increasing; (iii) when c0 < c < c1, the bifurcation curve is S-shaped; (iv) when c ≥ c1, the bifurcation curve is ⊂-shaped. This work is a continuation of the work by Liang and Wang [8] where authors studied this problem for α ≥ α1, and our results partially prove a conjecture on this problem for 4 ≤ α < α1 in [8].
Description
Keywords
Multiplicity, Positive solutions, Perturbed Gelfand equation, S-shaped bifurcation curve, C-shaped bifurcation curve, Time map
Citation
Liang, Y. H., & Wang, S. H. (2017). Classification and evolution of bifurcation curves for the one-dimensional perturbed Gelfand equation with mixed boundary conditions II. <i>Electronic Journal of Differential Equations, 2017</i>(61), pp. 1-12.
Rights
Attribution 4.0 International