Convexity for free boundaries with singular term (nonlinear elliptic case)

We consider a free boundary problem in an exterior domain \begin{cases}\begin{array}{cc} Lu=g(u) & $\text{in }$Ω\setminus K, \ u=1 & $\text{on }$\partial K,\ |\nabla u|=0 &$\text{on }$\partial Ω, \end{array}\end{cases} where $K$ is a (given) convex and compact set in ${\mathbb{R}}^n$ ($n\ge 2$), $\mathrm{\Omega }=\{u>0\}\supset K$ is an unknown set, and $L$ is either a fully nonlinear or the $p$-Laplace operator. Under suitable assumptions on $K$ and $g$, we prove the existence of a nonnegative quasi-concave solution to the above problem. We also consider the cases when the set $K$ is contained in $\{x_n=0\}$, and obtain similar results.