Entire subsolutions of a kind of k-Hessian type equations with gradient terms

In this paper, we consider a kind of $k$-Hessian type equations $S_k^{\frac{1}{k}}(D^{2}u+\mu \mathrm{\mid }Du\mathrm{\mid }I)=f(u)$ in ${\mathbb{R}}^n$, and provide a necessary and sufficient condition of $f$ on the existence and nonexistence of entire admissible subsolutions, which can be regarded as a generalized Keller-Osserman condition. The existence and nonexistence results are proved in different ranges of the parameter $\mu$ respectively, which embrace the standard Hessian equation case ($\mu =0$) by Ji and Bao (Proc Amer Math Soc 138: 175–188, 2010) as a typical example. The difference between the semilinear case ($k=1$) and the fully nonlinear case ($k\ge 2$) is also concerned.