A simplified proof of a conjecture for the perturbed Gelfand equation from combustion theory

For the perturbed Gelfand's equation on the unit ball in two dimensions, Y. Du and Y. Lou [5] proved that the curve of positive solutions is exactly S-shaped, for sufficiently small values of the secondary parameter. We present a simplified proof and some extensions. This problem is prominent in combustion theory, see e.g., the book of J. Bebernes and D. Eberly [1].