Maximal solutions for the ∞-eigenvalue problem

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

Advances in Calculus of Variations

De Gruyter

Año: 2017 vol. 0

1864-8258

In this article we prove that the first eigenvalue of the $infty-$Laplacian$$left{egin{array}{rclcl} min{ -Delta_infty v,, |abla v|-lambda_{1, infty}(Omega) v } & = & 0 & ext{in} & Omega \ v & = & 0 & ext{on} & partial Omega,end{array}ight.$$has a unique (up to scalar multiplication) maximal solution.This maximal solution can be obtained as the limit as $ell earrow 1$ of concave problems of the form$$left{egin{array}{rclcl} min{ -Delta_infty v_{ell},, |abla v_{ell}|-lambda_{1, infty}(Omega) v_{ell}^{ell} } & = & 0 & ext{in} & Omega \ v_{ell} & = & 0 & ext{on} & partial Omega.end{array}ight.$$In this way we obtain that the maximal eigenfunction is the unique one that is the limit of the concave problemsas happens for the usual eigenvalue problem for the $p-$Laplacian for a fixed $1