Journal
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS
Volume 73, Issue 7, Pages 2027-2035Publisher
PERGAMON-ELSEVIER SCIENCE LTD
DOI: 10.1016/j.na.2010.05.032
Keywords
infinity-Laplacian; p(x)-Laplacian; Dirichlet boundary conditions
Categories
Funding
- Fundacao para a Ciencia e a Tecnologia (Portugal)
- UBA [X066]
- CONICET (Argentina)
- [MTM2008-06326-C02-02]
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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.
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