Non-linear Dynamics Commons^™

The Gelfand Problem For The Infinity Laplacian, Fernando Charro, Byungjae Son, Peiyong Wang

Mathematics Faculty Research Publications

We study the asymptotic behavior as p → ∞ of the Gelfand problem

−Δ_pu = λe^u 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|−Λe^u, −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.