Existence and nonexistence of subsolutions for augmented Hessian equations

In this paper, we consider the augmented Hessian equations $ S_k^{\frac{1}{k}}[D^2u+\sigma(x)I] = f(u) $ in $ \mathbb{R}^{n} $ or $ \mathbb{R}^{n}_+ $. We first give the necessary and sufficient condition of the existence of classical subsolutions to the equations in $ \mathbb{R}^{n} $ for $ \sigma(x) = \alpha $, which is an extended Keller-Osserman condition. Then we obtain the nonexistence of positive viscosity subsolutions of the equations in $ \mathbb{R}^{n} $ or $ \mathbb{R}^{n}_+ $ for $ f(u) = u^p $ with $ p>1 $.