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

In this paper we study the behaviour of the solutions to the eigenvalue problem corresponding to the p(x)-Laplacian operator {-div(vertical bar del u vertical bar(p(x)-2)del u) = Lambda(p(x))vertical bar u vertical bar(p(x)-2)u, in Omega, u = 0, on theta Omega, as p(x) -> infinity. We consider a sequence of functions p(n)(x) that goes to infinity uniformly in (Omega) over bar. Under adequate hypotheses on the sequence P-n, namely that the limits del In p(n)(x) -> xi(x), and Pn/n (x) -> q(x) exist, we prove that the corresponding eigenvalues u(pn) and eigenfunctions up, verify that (Lambda p(n))(1/n) -> Lambda proportional to, u(pn) - u(proportional to) uniformly in (Omega) over bar. where Lambda(infinity), u(infinity), is a nontrivial viscosity solution of the following problem {min{-Delta(infinity) u(infinity). vertical bar del u(infinity)vertical bar(2) log(vertical bar del u infinity vertical bar)(xi, del u(infinity)) vertical bar del u(infinity)vertical bar(q) - Lambda(infinity)u(infinity)(q)} = 0, in Omega, u infinity = 0, on theta Omega. (C) 2009 Elsevier Inc. All rights reserved.