Existence and nonexistence of subsolutions for augmented Hessian equations

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