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

We study the Dirichlet problem $-÷(\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{p(x)-2}\mathrm{\nabla }u)=0$ in $\mathrm{\Omega }$, with $u=f$ on $\mathrm{\partial }\mathrm{\Omega }$ and $p(x)=\mathrm{\infty }$ in $D$, a subdomain of the reference domain $\mathrm{\Omega }$. The main issue is to give a proper sense to what a solution is. To this end, we consider the limit as $n \to \infty$ofthesolutions$u_n$tothecorrespondingproblemwhen$p_n(x) =p(x) \wedge n$,inparticular,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, $\mathrm{\infty }$-harmonic within $D$. Moreover, we examine this limit in the viscosity sense and find the boundary value problem it satisfies in the whole of $\mathrm{\Omega }$.