Limits as p(x)→∞p(x)→∞ of p(x)p(x)-harmonic functions

In this note we study the limit as p(x)→∞p(x)→∞ of solutions to −Δp(x)u=0−Δp(x)u=0 in a domain ΩΩ, with Dirichlet boundary conditions. Our approach consists in considering sequences of variable exponents converging uniformly to +∞+∞ and analyzing how the corresponding solutions of the problem converge and which equation is satisfied by the limit.