A singular perturbation problem for a quasilinear operator satisfying the natural growth condition of Lieberman

Abstract

In this paper we study the following problem. For any $\ep>0$, take $u^{\ep}$ a solution of, $$ Łu^{\ep}:= {div}\Big(\di\frac {g(|\nabla \uep|)}{|\nabla \uep|}\nabla \uep\Big)=β_{\ep}(u^{\ep}),\quad u^{\ep}\geq 0. $$ A solution to $(P_{\ep})$ is a function $u^{\ep}\in W^{1,G}(Ω)\cap L^{\infty}(Ω)$ such that $$ \int_Ω g(|\nabla u^{\ep}|) \frac{\nabla u^{\ep}}{|\nabla u^{\ep}|} \nabla ϕdx =-\int_Ω ϕβ_{\ep}(u^{\ep}) dx $$ for every $\varphi \in C_0^{\infty }(\Omega )$. Here $β_{\ep}(s)= \frac{1}{\ep} β(\frac{s}{\ep}), $ with $\beta \in Lip(\mathbb{R})$, $\beta >0$ in $(0,1)$ and $\beta =0$ otherwise. We are interested in the limiting problem, when $\ep\to 0$. As in previous work with $Ł=\Delta$ or $Ł={\Delta }_p$ we prove, under appropriate assumptions, that any limiting function is a weak solution to a free boundary problem. Moreover, for nondegenerate limits we prove that the reduced free boundary is a $C^{1,\alpha }$ surface. This result is new even for ${\Delta }_p$. Throughout the paper we assume that $g$ satisfies the conditions introduced by G. Lieberman in \cite{Li1}