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 (Formula Presented) where 4 ≤ a < a <sub>1</sub> ≈ 4.107. We prove that, for 4 ≤ a < a <sub>1</sub> , there exist two nonnegative c <sub>0</sub> = c <sub>0</sub> (a) < c <sub>1</sub> = c <sub>1</sub> (a) satisfying c <sub>0</sub> > 0 for 4 ≤ a < a <sup>∗</sup> ≈ 4.069, and c <sub>0</sub> = 0 for a <sup>∗</sup> ≤ a < a <sub>1</sub> , such that, on the (λ, ‖u‖ <sub>∞</sub> )-plane, (i) when 0 < c < c <sub>0</sub> , the bifurcation curve is strictly increasing; (ii) when c = c <sub>0</sub> , the bifurcation curve is monotone increasing; (iii) when c <sub>0</sub> < c < c <sub>1</sub> , the bifurcation curve is S-shaped; (iv) when c ≥ c <sub>1</sub> , the bifurcation curve is ⊂-shaped. This work is a continuation of the work by Liang and Wang [8] where authors studied this problem for a ≥ a <sub>1</sub> , and our results partially prove a conjecture on this problem for 4 ≤ a < a <sub>1</sub> in [8].