Classification and evolution of bifurcation curves for the one-dimensional perturbed Gelfand equation with mixed boundary conditions

Abstract

We study the classification and evolution of bifurcation curves of positive solutions for the onedimensional perturbed Gelfand equation with mixed boundary conditions given by {u"(x) + lambda exp(au/a+u) = 0, 0 < x < 1, u(0) = 0, u\' (1) = -c < 0. We prove that, for positive a <= a(0) (approximate to 0.501) and c > 0, the bifurcation curve is strictly increasing on the (A, II ulloo)-plane, and there exists a positive Xo such that the problem has no positive solution for 0 < lambda < lambda(0) and exactly one positive solution for lambda > lambda(0). While for a >= a(1) (approximate to 4.107), there exists c(1) (= c(1) (a)) > 1.057 such that, on the (A, 11u1100) -plane, (i) when 0 < c < cl, the bifurcation curve is S-shaped, and the problem has at least three positive solutions for some range of positive A; (ii) when c > cl, the bifurcation curve is c-shaped and the problem has at least two positive solutions for some range of positive A. (C) 2016 Elsevier Inc. All rights reserved.