The $\infty$-eigenvalue problem with a sign-changing weight

Abstract

Let Ω ⊂ R n be a smooth bounded domain and m ∈ C(Ω) be a sign-changing weight function. For 1 < p < ∞, consider the eigenvalue problem −∆pu = λm(x)|u| p−2 u in Ω, u = 0 on ∂Ω, where ∆pu is the usual p-Laplacian. Our purpose in this article is to study the limit as p → ∞ for the eigenvalues λ k,p (m) of the aforementioned problem. In addition, we describe the limit of some normalized associated eigenfunctions when k = 1.