On the solutions to p-Poisson equation with Robin boundary conditions when p goes to +∞

Abstract

We study the behaviour, when p → + ∞ p\to +\infty , of the first p-Laplacian eigenvalues with Robin boundary conditions and the limit of the associated eigenfunctions. We prove that the limit of the eigenfunctions is a viscosity solution to an eigenvalue problem for the so-called ∞ \infty -Laplacian. Moreover, in the second part of the article, we focus our attention on the p-Poisson equation when the datum f f belongs to L ∞ ( Ω ) {L}^{\infty }\left(\Omega ) and we study the behaviour of solutions when p → ∞ p\to \infty .