Liouville theorems for elliptic systems with nonlinear gradient terms

Abstract

In this paper we obtain Liouville type theorems for positive supersolutions of the elliptic system{−Δu+|∇u|q=λf(v)−Δv+|∇v|q=μg(u) in exterior domains of RN, where q>1 and the functions f and g behave like a power near zero or infinity. We show that positive supersolutions do not exist in some ranges of the parameters involved, which are optimal in the special case where f(v)=vp and g(u)=us, with p,s>0.