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In this paper we study the following problem. For any $\ep>0$, take $u^{\ep}$a solution of, $$  \L u^{\ep}:= {div}\Big(\di\frac {g(|\nabla \uep|)}{|\nabla \uep|}\nabla\uep\Big)=\beta_{\ep}(u^{\ep}),\quad u^{\ep}\geq 0. $$  A solution to $(P_{\ep})$ is a function $u^{\ep}\in W^{1,G}(\Omega)\capL^{\infty}(\Omega)$ such that $$ \int_{\Omega} g(|\nabla u^{\ep}|) \frac{\nablau^{\ep}}{|\nabla u^{\ep}|} \nabla \phi dx =-\int_{\Omega} \phi\beta_{\ep}(u^{\ep}) dx $$ for every $\phi \in C_0^{\infty}(\Omega)$.  Here $\beta_{\ep}(s)= \frac{1}{\ep} \beta(\frac{s}{\ep}), $ with $\beta\in{Lip}(\R)$, $\beta>0$ in $(0,1)$ and $\beta=0$ otherwise.  We are interested in the limiting problem, when $\ep\to 0$. As in previouswork with $\L=\Delta$ or $\L=\Delta_p$ 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 introducedby G. Lieberman in \cite{Li1}

A Singular Perturbation Problem for a Quasi-Linear Operator Satisfying the Natural Growth Condition of Lieberman