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

Abstract

In this paper we study the following singular perturbation problem for the $p_{\epsilon }(x)$-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 $\epsilon >0$, ${\beta }_{\epsilon }(s)=\frac{1}{\epsilon }\beta (\frac{s}{\epsilon })$, with $\beta$ a Lipschitz function satisfying $\beta >0$ in $(0,1)$, $\beta \equiv 0$ outside $(0,1)$ and $\int \beta (s)ds=M$. The functions $u^{\epsilon }$, $f_{\epsilon }$ and $p_{\epsilon }$ 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 \[ \begin{cases} Δ_{p(x)}u= f & \mbox{in }\{u>0\}\\ u=0,\ |\nabla u| = λ^*(x) & \mbox{on }\partial\{u>0\} \end{cases} \] with ${\lambda }^{\ast }(x)=(\frac{p(x)}{p(x)-1}M{)}^{1/p(x)}$, $p=\lim p_{\epsilon }$ and $f=\lim f_{\epsilon }$. 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.

Nonlinear Analysis TM&A, to appear