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

In this paper we study the following singular perturbation problem for the $p_{\epsilon }(x)$-Laplacian: $${\mathrm{\Delta }}_{p_{\epsilon }(x)}{u}^{\epsilon }:=div(\mathrm{\mid }\mathrm{\nabla }{u}^{\epsilon }(x){\mathrm{\mid }}^{p_{\epsilon }(x)-2}\mathrm{\nabla }{u}^{\epsilon })={\beta }_{\epsilon }(u^{\epsilon })+f_{\epsilon },u^{\epsilon }\ge 0,$$ where $\epsilon >0$, ${\beta}{\varepsilon}(s)={1 \over \varepsilon} \beta({s \over \varepsilon})$,with$\beta$aLipschitzfunctionsatisfying$\beta>0$in$(0,1)$,$\beta\equiv 0$outside$(0,1)$and$\int \beta(s), ds=M$\mathrm{.}Thefunctions$u^\varepsilon$,$f\varepsilon$and$p_\varepsilon$ are uniformly bounded. We prove uniform Lipschitz regularity, we pass to the limit $(\epsilon \to 0)$ and we show that, under suitable assumptions, limit functions are weak solutions to the free boundary problem: $u\ge 0$ and $$\{\begin{array}{ll}{\mathrm{\Delta }}_{p(x)}u=f& in\{u>0\}\\ u=0,\mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid }={\lambda }^{\ast }(x)& on\mathrm{\partial }\{u>0\}\end{array}$$ with ${\lambda }^{\ast }(x)=(\frac{p(x)}{p(x)-1}M{)}^{1\mathrm{/}p(x)}$, $p=\lim p_\varepsilon$and$f=\lim f_\varepsilon$. In \cite{LW4} we prove that the free boundary of a weak solution is a $C^{1,\alpha }$ surface near flat free boundary points. This result applies, in particular, to the limit functions studied in this paper.