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

In this paper we study the following singular perturbation problem for the $p_\varepsilon(x)$-Laplacian: \[ Δ_{p_\varepsilon(x)}u^\varepsilon:=\mbox{div}(|\nabla u^\varepsilon(x)|^{p_\varepsilon(x)-2}\nabla u^\varepsilon)=β_{\varepsilon}(u^\varepsilon)+f_\varepsilon, \quad u^\varepsilon\geq 0, \] where $\varepsilon>0$, $β_{\varepsilon}(s)={1 \over \varepsilon} β({s \over \varepsilon})$, with $β$ a Lipschitz function satisfying $β>0$ in $(0,1)$, $β\equiv 0$ outside $(0,1)$ and $\int β(s)\, ds=M$. The functions $u^\varepsilon$, $f_\varepsilon$ and $p_\varepsilon$ are uniformly bounded. We prove uniform Lipschitz regularity, we pass to the limit $(\varepsilon\to 0)$ and we show that, under suitable assumptions, limit functions are weak solutions to the free boundary problem: $u\ge0$ and \[ \begin{cases} Δ_{p(x)}u= f & \mbox{in }\{u>0\}\\ u=0,\ |\nabla u| = λ^*(x) & \mbox{on }\partial\{u>0\} \end{cases} \] with $λ^*(x)=\Big(\frac{p(x)}{p(x)-1}\,M\Big)^{1/p(x)}$, $p=\lim p_\varepsilon$ and $f=\lim f_\varepsilon$. In \cite{LW4} we prove that the free boundary of a weak solution is a $C^{1,α}$ surface near flat free boundary points. This result applies, in particular, to the limit functions studied in this paper.