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

We study the Dirichlet problem - div (| ∇ u |p (x) - 2 ∇ u) = 0 in Ω, with u = f on ∂Ω and p (x) = ∞ in D, a subdomain of the reference domain Ω. The main issue is to give a proper sense to what a solution is. To this end, we consider the limit as n → ∞ of the solutions un to the corresponding problem when pn (x) = p (x) ∧ n, in particular, with pn = 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, ∞-harmonic within D. Moreover, we examine this limit in the viscosity sense and find the boundary value problem it satisfies in the whole of Ω. © 2009 Elsevier Masson SAS. All rights reserved.