An inhomogeneous singular perturbation problem for the p(x)-Laplacian

• Published in: Nonlinear analysis 2016-06, Vol.138, p.300-325
• Main Authors: Lederman, Claudia, Wolanski, Noemi
• Format: Article
• Language: English
• Subjects: Free boundary problem; Singular perturbation; Variable exponent spaces
• ISSN: 0362-546X; 1873-5215
• DOI: 10.1016/j.na.2015.09.026

In this paper we study the following singular perturbation problem for the pε(x)-Laplacian: (Pε(fε,pε))Δpε(x)uε:=div(|∇uε(x)|pε(x)−2∇uε)=βε(uε)+fε,uε≥0, where ε>0, βε(s)=1εβ(sε), with β a Lipschitz function satisfying β>0 in (0,1), β≡0 outside (0,1) and ∫β(s)ds=M. The functions uε, fε and pε are uniformly bounded. We prove uniform Lipschitz regularity, we pass to the limit (ε→0) and we show that, under suitable assumptions, limit functions are weak solutions to the free boundary problem: u≥0 and (P(f,p,λ∗)){Δp(x)u=fin  {u>0}u=0,|∇u|=λ∗(x)on  ∂{u>0} with λ∗(x)=(p(x)p(x)−1M)1/p(x), p=limpε and f=limfε. In Lederman and Wolanski (submitted) we prove that the free boundary of a weak solution is a C1,α surface near flat free boundary points. This result applies, in particular, to the limit functions studied in this paper.