The ∞ -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 {-Δpu=λm(x)|u|p-2uinΩ,u=0on∂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.