Maximal solutions for the ∞-eigenvalue problem

ROSSI, JULIO D.; SALORT, ARIEL M.; DA SILVA, JOÃO VITOR

Advances in Calculus of Variations

De Gruyter

Año: 2019 vol. 12 p. 181 - 191

1864-8258

In this article we prove that the first eigenvalue of the ∞-Laplacian{min{−Δ∞v,|∇v|−λ1,∞(Ω)v}v=0=0in Ω,on ∂Ω,has a unique (up to scalar multiplication) maximal solution. This maximal solution can be obtained as the limit as ℓ↗1 of concave problems of the form{min{−Δ∞vℓ,|∇vℓ|−λ1,∞(Ω)vℓℓ}vℓ=0=0in Ω,on ∂Ω.In this way we obtain that the maximal eigenfunction is the unique one that is the limit of the sub-homogeneous problems as happens for the usual eigenvalue problem for the p-Laplacian for a fixed 1