  In this paper, we discuss the more general Hessian inequality
$\sigma_{k}^{\frac{1}{k}}(\lambda (D_i (A\left(|Du|\right) D_j u)))\geq f(u)$
including the Laplacian, p-Laplacian, mean curvature, Hessian, k-mean curvature
operators, and provide a necessary and sufficient condition on the global
solvability, which can be regarded as generalized Keller-Osserman conditions.
