  In this paper we develop a new theory for the existence, localization and
multiplicity of positive solutions for a class of non-variational,quasilinear,
elliptic systems. In order to do this, we provide a fairly general abstract
framework for the existence of fixed points of nonlinear operators acting on
cones that satisfy an inequality of Harnack type. Our methodology relies on
fixed point index theory. We also provide a non-existence result and an example
to illustrate the theory.
