The behaviour of the p (x)-Laplacian eigenvalue problem as p (x) → ∞ - Pérez-Llanos, M. (2010)
In this paper we study the behaviour of the solutions to the eigenvalue problem corresponding to the p (x)-Laplacian operator{(- div (| ∇ u |p (x) - 2 ∇ u) = Λp (x) | u |p (x) - 2 u,, in Ω,; u = 0,, on ∂ Ω,) as p (x) → ∞. We consider a sequence of functions pn (x) that goes to infinity uniformly in over(Ω, -). Under adequate hypotheses on the sequence pn, namely that the limits∇ ln pn (x) → ξ (x), and frac(pn, n) (x) → q (x) exist, we prove that the corresponding eigenvalues Λpn and eigenfunctions upn verify that(Λpn)1 / n → Λ∞, upn → u∞ uniformly in over(Ω, -), where Λ∞, u∞ is a nontrivial viscosity solution of the following problem{(min {- Δ∞ u∞ - | ∇ u∞ |2 log (| ∇ u∞ |) 〈 ξ, ∇ u∞ 〉, | ∇ u∞ |q - Λ∞ u∞ q} = 0, in Ω,; u∞ = 0, on ∂ Ω .). © 2009 Elsevier Inc. All rights reserved.