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

In this paper we study the following problem. For any $\text{ep}>0$, take $u^{\text{ep}}$ a solution of, $$\text{L}{u}^{\text{ep}}:=div\left(\text{di}\frac{g(\mathrm{\mid }\mathrm{\nabla }\text{uep}\mathrm{\mid })}{\mathrm{\mid }\mathrm{\nabla }\text{uep}\mathrm{\mid }}\mathrm{\nabla }\text{uep}\right)={\beta }_{\text{ep}}(u^{\text{ep}}),u^{\text{ep}}\ge 0.$$ A solution to $(P_{\text{ep}})$ is a function $u^{\ep}\in W^{1,G}(\Omega)\cap L^{\infty}(\Omega)$suchthat$$ \int_{\Omega} g(|\nabla u^{\ep}|) \frac{\nabla u^{\ep}}{|\nabla u^{\ep}|} \nabla \phi dx =-\int_{\Omega} \phi \beta_{\ep}(u^{\ep}) dx $$forevery$\phi \in C_0^{\infty}(\Omega)$. Here ${\beta }_{\text{ep}}(s)=\frac{1}{\text{ep}}\beta (\frac{s}{\text{ep}}),$ with $\beta\in {Lip}(\R)$,$\beta>0$in$(0,1)$and$\beta=0$ otherwise. We are interested in the limiting problem, when $\text{ep}\to 0$. As in previous work with $\text{L}=\mathrm{\Delta }$ or $\text{L}={\mathrm{\Delta }}_p$ 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 ${\mathrm{\Delta }}_p$. Throughout the paper we assume that $g$ satisfies the conditions introduced by G. Lieberman in \cite{Li1}