Proof of a Conjecture for the Perturbed Gelfand Equation from Combustion Theory

In this paper, we prove that when the domain B is a two dimensional ball, for all small positive , the solution curve {(λ, u)} of the perturbed Gelfand equation −Δu=λeu/(1+ u) with u∂B=0 is exactly S-shaped. This settles a long-standing conjecture on the exact number of solutions to this equation