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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 $D$, asubdomain of the reference domain $\Omega$. The main issue is to give a propersense 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 onthe data, we find that such a limit exists and that it can be characterized asthe unique solution of a variational minimization problem which is, inaddition, $\infty$-harmonic within $D$. Moreover, we examine this limit in theviscosity sense and find the boundary value problem it satisfies in the wholeof $\Omega$.

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