The Gelfand problem for the Infinity Laplacian

2021/12/16 by Charro, Fernando, Son, Byungjae, Wang, Peiyong
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2112.09247

Abstract

We study the asymptotic behavior as p→∞ of the Gelfand problem -Δ_p u=λ e^u \textrmin Ω⊂ℝⁿ, u=0 \textrmon ∂Ω. Under an appropriate rescaling on u and λ, we prove uniform convergence of solutions of the Gelfand problem to solutions of min\|∇u|-Λ e^u, -Δ_∞u\=0 \textrmin Ω, u=0 on ∂Ω. We discuss existence, non-existence, and multiplicity of solutions of the limit problem in terms of Λ.

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