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In this paper, we consider a kind of $k$-Hessian type equations$S_k^{\frac{1}{k}}(D^2u+\mu|D u|I)= f(u)$ in $\mathbb{R}^n$, and provide anecessary and sufficient condition of $f$ on the existence and nonexistence ofentire admissible subsolutions, which can be regarded as a generalizedKeller-Osserman condition. The existence and nonexistence results are proved indifferent ranges of the parameter $\mu$ respectively, which embrace thestandard 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 semilinearcase ($k=1$) and the fully nonlinear case ($k\ge 2$) is also concerned.

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