Abstract
We study the classification and evolution of bifurcation curves of positive solutions for the one-dimensional perturbed Gelfand equation with mixed boundary conditions given by {u″ (x) + λ exp (au/a+u) = 0,0 < x < 1, u(0) = 0, u'(1) = -c < 0. We prove that, for positive a ≤ a <sub>0</sub> (≈ 0.501) and c > 0, the bifurcation curve is strictly increasing on the (λ, u <sub>∞</sub> )-plane, and there exists a positive λ <sub>0</sub> such that the problem has no positive solution for 0 < λ < λ <sub>0</sub> and exactly one positive solution for λ ≥ λ <sub>0</sub> . While for a ≥ a <sub>1</sub> (≈ 4.107), there exists c <sub>1</sub> (= c <sub>1</sub> (a)) > 1.057 such that, on the (λ, u <sub>∞</sub> )-plane, (i) when 0 < c < c <sub>1</sub> , the bifurcation curve is S-shaped, and the problem has at least three positive solutions for some range of positive λ (ii) when c ≥ c <sub>1</sub> , the bifurcation curve is ⊂-shaped and the problem has at least two positive solutions for some range of positive λ.