Solvability of nonlinear elliptic equations with gradient terms

We study the solvability in the whole Euclidean space of coercive quasi-linear and fully nonlinear elliptic equations modeled on ∆u±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 R . 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.