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

In this paper we study the following singular perturbation problem for the pε(x)pε(x)-Laplacian: equation(Pε(fε,pε)Pε(fε,pε))Δpε(x)uε:=div(|∇uε(x)|pε(x)−2∇uε)=βε(uε)+fε,uε≥0, where ε>0ε>0, βε(s)=1εβ(sε), with ββ a Lipschitz function satisfying β>0β>0 in (0,1)(0,1), β≡0β≡0 outside (0,1)(0,1) and ∫β(s)ds=M. The functions uεuε, fεfε and pεpε are uniformly bounded. We prove uniform Lipschitz regularity, we pass to the limit (ε→0)(ε→0) and we show that, under suitable assumptions, limit functions are weak solutions to the free boundary problem: u≥0u≥0 and equation(P(f,p,λ∗)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=limpεp=limpε and f=limfεf=limfε.In Lederman and Wolanski (submitted) we prove that the free boundary of a weak solution is a C1,αC1,α surface near flat free boundary points. This result applies, in particular, to the limit functions studied in this paper.