A singular perturbation problem for the p(x)-Laplacian

Abstract

We present results for the following singular perturbation problem: ∆p(x)uε := div(|∇uε(x)| p(x)−2∇uε) = βε(uε) + f ε, uε ≥ 0 (Pε(f ε)) in Ω ⊂ RN , where ε > 0, βε(s) = 1 εβ( s ε ), with β a Lipschitz function satisfying β > 0 in (0, 1), β ≡ 0 outside (0, 1) and β(s) ds = M. The functions uε and f ε are uniformly bounded. We prove uniform Lipschitz regularity, we pass to the limit (ε → 0) and we show that limit functions are weak solutions to a free boundary problem.