The behaviour of the p(x)p(x)-Laplacian eigenvalue problem as p(x)→∞p(x)→∞

In this paper we study the behaviour of the solutions to the eigenvalue problem corresponding to the p(x)p(x)-Laplacian operator{−div(|∇u|p(x)−2∇u)=Λp(x)|u|p(x)−2u,in Ω,u=0,on ∂Ω, as p(x)→∞p(x)→∞. We consider a sequence of functions pn(x)pn(x) that goes to infinity uniformly in Ω¯. Under adequate hypotheses on the sequence pnpn, namely that the limits∇lnpn(x)→ξ(x),andpnn(x)→q(x) exist, we prove that the corresponding eigenvalues ΛpnΛpn and eigenfunctions upnupn verify that(Λpn)1/n→Λ∞,upn→u∞uniformly in Ω¯, where Λ∞Λ∞, u∞u∞ is a nontrivial viscosity solution of the following problem{min{−Δ∞u∞−|∇u∞|2log(|∇u∞|)〈ξ,∇u∞〉,|∇u∞|q−Λ∞u∞q}=0,in Ω,u∞=0,on ∂Ω.