Classification of supersolutions and Liouville theorems for some nonlinear elliptic problems

In this paper we consider positive supersolutions of the elliptic equation - u =f(u) |∇ u|q, posed in exterior domains of RN (N≥ 2), where f iscontinuous in [0,+∞) and positive in (0,+∞) and $q>0$. We classifysupersolutions u into four types depending on the function m(R)=inf|x|=Ru(x) for large R, and give necessary and sufficient conditions in order to havesupersolutions of each of these types. As a consequence, we also obtain Liouvilletheorems for supersolutions depending on the values of N, q and on someintegrability properties on f at zero or infinity. We also describe these questions whenthe equation is posed in the whole RN.