Nonexistence of positive supersolutions to some nonlinear elliptic problems

In this paper we obtain Liouville type theorems for positive supersolutions of the elliptic problem -Delta u + vertical bar del u vertical bar(q) = lambda f(u) in exterior domains of R-N. 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) = s(p). The related problem -Delta u - vertical bar del u vertical bar(q) = f(u) is also analyzed. (C) 2012 Elsevier Masson SAS. All rights reserved.