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 lim inf x∈Ω [x] → ∞ u(x) +1 / dist (x, ∂Ω) = 0 and then, in particular, for strictly sublinear super-solutions in a domain σ 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.