Harnack's inequality for p(⋅)-harmonic functions with unbounded exponent p

We study properties of the function u=limλ→∞uλ, where uλ is the solution of the min{p(⋅),λ}-Laplacian Dirichlet problem with bounded Sobolev boundary function. Here is a variable exponent such that 1/p is Lipschitz continuous. We derive Bloch-type estimates and using them we prove Harnack's inequality in cases of unbounded but finite exponent.