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

Abstract

We consider a free boundary problem in an exterior domain $$\begin{aligned} {\left\{ \begin{array}{ll} Lu=g(u)&{}\text {in }\Omega \setminus K,\\ u=1 &{} \text {on }\partial K,\\ |\nabla u|=0 &{}\text {on }\partial \Omega , \end{array}\right. } \end{aligned}$$ L u = g ( u ) in Ω \ K , u = 1 on ∂ K , | ∇ u | = 0 on ∂ Ω , where K is a (given) convex and compact set in $${\mathbb R}^n$$ R n ($$n\ge 2$$ n ≥ 2 ), $$\Omega =\{u>0\}\supset K$$ 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 $$\{x_n=0\}$$ { x n = 0 } , and obtain similar results.