Non-existence of Positive Solutions of Fully Nonlinear Elliptic Equations in Unbounded Domains

In this paper we consider fully nonlinear elliptic operators of the form F (x, u,Du,D2u). Our aim is to prove that, under suitable assumptions on F , the only nonnegative viscosity super-solution u of F (x, u,Du,D2u) = 0 in an unbounded domain Ω is u ≡ 0. We show that this uniqueness result holds for the class of nonnegative super-solutions u satisfying inf x∈Ω u(x) + 1 dist(x, ∂Ω) = 0, and then, in particular, for strictly sublinear super-solutions in a domain Ω containing an open cone. In the special case that Ω = RN , or that F is the Bellman operator, we show that the same result holds for the whole class of nonnegative super-solutions. Our principal assumption on the operator F involves its zero and first order dependence when |x| → ∞. The same kind of assumption was introduced in a recent paper in collaboration with H. Berestycki and F. Hamel [4] to establish a Liouville type result for semilinear equations. The strategy we follow to prove our main results is the same as in [4], even if here we consider fully nonlinear operators, possibly unbounded solutions and more general domains.