Liang, Yu-HaoWang, Shin-Hwa2022-04-012022-04-012017-02-28Liang, 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. 1-12.1072-6691https://hdl.handle.net/10877/15587In 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].Text12 pages1 file (.pdf)enAttribution 4.0 InternationalMultiplicityPositive solutionsPerturbed Gelfand equationS-shaped bifurcation curveC-shaped bifurcation curveTime mapArticle