Limit as $p(x)\to \mathrm{\infty }$ of $p(x)$-harmonic functions for unbounded $p(x)$

It is shown that if $p_j$ is a sequence of continuous, unbounded exponents on a bounded, Lipschitz domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$ with $1<\underset{\in \mathrm{\Omega }}{inf}{p}_j(x)$ and $p_j\to \mathrm{\infty }$ uniformly, then the sequence $(u_j)$ of solutions of the $p_j(\cdot )$-Laplacian converges to the viscosity solution of a suitable differential operator. The novelty here is that each term of the sequence of exponents $(p_j)$ is allowed to be unbounded in $\mathrm{\Omega }$.