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

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