A singular perturbation problem for the p(x)-Laplacian

LEDERMAN, CLAUDIA; WOLANSKI, NOEMÍ

Matemática Aplicada Computacional e Industrial

Asociación Argentina de Matemática Aplicada, Computacional e Industrial

Lugar: Santa Fé; Año: 2013 vol. 4 p. 485 - 488

2314-3282

We present results for the following   singular perturbation  problem: \begin{equation} \label{eq}\tag{$P_\ep(\fep)$} \Delta_{p(x)}\uep:=\mbox{div}(|\nabla \uep(x)|^{p(x)-2}\nabla \uep)={\beta}_{\varepsilon}(\uep)+\fep, \quad u^{\ep}\geq 0 \end{equation} in $\Omega\subset \Bbb R^{N}$, 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 $\uep$ and $\fep$ are uniformly  bounded. We prove uniform Lipschitz regularity, we pass to the  limit $(\ep\to 0)$ and we show that limit functions are weak  solutions to a free boundary problem.