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

We study the Dirichlet problem $-÷(|\nabla u|^{p(x)-2} \nabla u) =0$ in $Ω$, with $u=f$ on $\partial Ω$ and $p(x) = \infty$ 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 $n \to \infty$ 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 $Ω$.

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