Entire solutions of the generalized Hessian inequality

Abstract

In this paper, we discuss the more general Hessian inequality ${\sigma }_k^{\frac{1}{k}}(\lambda (D_i(A\left(|Du|\right)D_{j}u)))\ge f(u)$ 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.