Maximal solutions for the ∞-eigenvalue problem

Abstract

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