  We study the Dirichlet problem $-\div(|\nabla u|^{p(x)-2} \nabla u) =0 $ in
$\Omega$, with $u=f$ on $\partial \Omega$ and $p(x) = \infty$ 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 $n \to
\infty$ of the solutions $u_n$ to the corresponding problem when $p_n(x) =p(x)
\wedge n$, 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, $\infty$-harmonic within $D$. Moreover, we examine this limit in the
viscosity sense and find the boundary value problem it satisfies in the whole
of $\Omega$.
