We study the asymptotic behavior as p → ∞ of the Gelfand problem
−Δpu = λeu in Ω ⊂ Rn, u = 0 on ∂Ω.
Under an appropriate rescaling on u and λ, we prove uniform convergence of solutions of the Gelfand problem to solutions of
min{|∇u|−Λeu, −Δ∞u} = 0 in Ω, u = 0 on ∂Ω.
We discuss existence, non-existence, and multiplicity of solutions of the limit problem in terms of Λ
