Classification and evolution of bifurcation curves for the one-dimensional perturbed Gelfand equation with mixed boundary conditions II
| dc.contributor.author | Liang, Yu-Hao | |
| dc.contributor.author | Wang, Shin-Hwa | |
| dc.date.accessioned | 2022-04-01T13:12:44Z | |
| dc.date.available | 2022-04-01T13:12:44Z | |
| dc.date.issued | 2017-02-28 | |
| dc.description.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]. | |
| dc.description.department | Mathematics | |
| dc.format | Text | |
| dc.format.extent | 12 pages | |
| dc.format.medium | 1 file (.pdf) | |
| dc.identifier.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. | |
| dc.identifier.issn | 1072-6691 | |
| dc.identifier.uri | https://hdl.handle.net/10877/15587 | |
| dc.language.iso | en | |
| dc.publisher | Texas State University, Department of Mathematics | |
| dc.rights | Attribution 4.0 International | |
| dc.rights.uri | https://creativecommons.org/licenses/by/4.0/ | |
| dc.source | Electronic Journal of Differential Equations, 2017, San Marcos, Texas: Texas State University and University of North Texas. | |
| dc.subject | Multiplicity | |
| dc.subject | Positive solutions | |
| dc.subject | Perturbed Gelfand equation | |
| dc.subject | S-shaped bifurcation curve | |
| dc.subject | C-shaped bifurcation curve | |
| dc.subject | Time map | |
| dc.title | Classification and evolution of bifurcation curves for the one-dimensional perturbed Gelfand equation with mixed boundary conditions II | en_US |
| dc.type | Article |