The ∞-eigenvalue problem with a sign-changing weight

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 - Δ p u = λ m ( x ) | u | p - 2 u in Ω , u = 0 on ∂ Ω , where Δ p u 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 .