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

We consider a free boundary problem in an exterior domain {[ Lu=g(u) in Ω∖ K,; u=1 on ∂ K,; |∇ u|=0 on ∂Ω , ]. where K is a (given) convex and compact set in ℝ^n ( n≥ 2 ), Ω ={u>0}⊃ 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.