Harnackʹs inequality for -harmonic functions with unbounded exponent p

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