The behaviour of the -Laplacian eigenvalue problem as

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 ) = Λ p ( x ) | u | p ( x ) − 2 u , in Ω , u = 0 , on ∂ Ω , as p ( x ) → ∞ . We consider a sequence of functions p n ( x ) that goes to infinity uniformly in Ω ¯ . Under adequate hypotheses on the sequence p n , namely that the limits ∇ ln p n ( x ) → ξ ( x ) , and p n n ( x ) → q ( x ) exist, we prove that the corresponding eigenvalues Λ p n and eigenfunctions u p n verify that ( Λ p n ) 1 / n → Λ ∞ , u p n → u ∞ uniformly in 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 .