Nonexistence of positive supersolutions to some nonlinear elliptic problems

Abstract

In this paper we obtain Liouville type theorems for positive supersolutions of the elliptic problem −Δu+|∇u|q=λf(u) in exterior domains of RN. Here q>1 and the function f can be compared with a power p near zero or infinity. We show that positive supersolutions do not exist in some ranges of the parameters p and q which turn out to be optimal for the model case f(s)=sp. The related problem −Δu−|∇u|q=f(u) is also analyzed.