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.