  We study a singularly perturbed problem related to infinity Laplacian
operator with prescribed boundary values in a region. We prove that solutions
are locally (uniformly) Lipschitz continuous, they grow as a linear function,
are strongly non-degenerate and have porous level surfaces. Moreover, for some
restricted cases we show the finiteness of the (n-1)-dimensional Hausdorff
measure of level sets. The analysis of the asymptotic limits is carried out as
well.
