In this paper, we discuss the solvability of the p-k-Hessian inequality $\sigma _{k}^{\frac 1k} ( \lambda ( D_{i} (|Du|^{p-2}$ $ D_{j}u ) ) ) \geq f(u)$ on the entire space $\mathbb {R}^{n}$ and provide a necessary and sufficient condition, which can be regarded as a generalized Keller–Osserman condition. Furthermore, we obtain the optimal regularity of solution.