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

LEDERMAN, CLAUDIA; WOLANSKI, NOEMÍ

JOURNAL OF NONLINEAR ANALYSIS

PERGAMON-ELSEVIER SCIENCE LTD

Lugar: Amsterdam; Año: 2016 vol. 138 p. 300 - 325

0362-546X

In this paper we study the following singular perturbation problem for the $p_\ep(x)$-Laplacian:\begin{equation}\label{eq}\tag{$P_\ep(\fep, p_\ep)$}\Delta_{p_\ep(x)}\uep:=\mbox{div}(|\nabla \uep(x)|^{p_\ep(x)-2}\nabla\uep)={\beta}_{\varepsilon}(\uep)+\fep, \quad u^{\ep}\geq 0,\end{equation} where $\ep>0$,${\beta}_{\varepsilon}(s)={1 \over \varepsilon} \beta({s \over\varepsilon})$, 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 $\uep$, $\fep$ and $p_\ep$ are uniformly bounded.We prove uniform Lipschitz regularity, we pass to the limit $(\ep\to 0)$ and we show that, under suitable assumptions, limit functions are weak solutions to the free boundary problem: $u\ge0$ and\begin{equation}\label{fbp-px}\tag{$P(f,p,{\lambda}^*)$}\begin{cases}\Delta_{p(x)}u= f & \mbox{in }\{u>0\}\\u=0,\ |\nabla u| = \lambda^*(x) & \mbox{on }\partial\{u>0\}\end{cases}\end{equation}with $\lambda^*(x)=\Big(\frac{p(x)}{p(x)-1}\,M\Big)^{1/p(x)}$,$p=\lim p_\ep$ and $f=\lim \fep$.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.