The limit as p(x)→ ∞ of solutions to the inhomogeneous Dirichlet problem of the p(x)-Laplacian

In this work, we study the behaviour of the solutions to the following Dirichlet problem related to the p(x)-Laplacian operator, { -div(vertical bar del u vertical bar(p(x)-2)del U) = f(x), in Omega u= 0, on partial derivative Omega, as p(x) -> infinity, for some suitable functions f. We consider a sequence of functions p(n)(x) that goes to infinity uniformly in (Omega) over bar. Under adequate hypotheses on the sequence p(n), basically, that the following two limits exist, max p(n) 3 lim del lnp(n)(x) = xi(x), and lim sup x epsilon(Omega) over bar /minp(n) <= k, for some k > 0, n ->infinity n ->infinity x epsilon(Omega) over bar we prove that u(pn) -> u(infinity) uniformly in (Omega) over bar. In addition, we find that u(infinity) solves a certain partial differential equation (PDE) problem ( that depends on f) in the viscosity sense. In particular, when f equivalent to 1 in Omega, we get u(infinity) (x) = dist(x, partial derivative Omega), and it turns out that the limit equation is vertical bar del u vertical bar = 1. (C) 2010 Elsevier Ltd. All rights reserved.