p(x)-Harmonic functions with unbounded exponent in a subdomain

We study the Dirichlet problem -div(vertical bar del u vertical bar(p(x)-2)del u) = 0 in Omega, with u = f on partial derivative Omega and p(x) = infinity in D, a subdomain of the reference domain Omega. The main issue is to give a proper sense to what a solution is. To this end, we consider the limit as n -> infinity of the solutions u(n) to the corresponding problem when p(n)(x) = p(x) boolean AND n, in particular, with p(n) = n in D. Under suitable assumptions on the data, we find that such a limit exists and that it can be characterized as the unique solution of a variational minimization problem which is, in addition, infinity-harmonic within D. Moreover, we examine this limit in the viscosity sense and find the boundary value problem it satisfies in the whole of Omega. (C) 2009 Elsevier Masson SAS. All rights reserved.