We study the solvability in the whole Euclidean space of coercive
quasi-linear and fully nonlinear elliptic equations modeled on Δu±g(∣∇u∣)=f(u), u≥0, where f and g are increasing continuous
functions. We give conditions on f and g which guarantee the availability
or the absence of positive solutions of such equations in RN. Our results
considerably improve the existing ones and are sharp or close to sharp in the
model cases. In particular, we completely characterize the solvability of such
equations when f and g have power growth at infinity. We also derive a
solvability statement for coercive equations in general form