  For the perturbed Gelfand's equation on the unit ball in two dimensions, Y.
Du and Y. Lou [4] 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].
