Necessary and sufficient conditions on entire solvability for real \((n-1)\) Monge–Ampère equation

In this paper, we establish a necessary and sufficient condition on solvability for the entire sub-solutions to the real \((n-1)\) $(n-1)$ Monge–Ampère equation \(\textrm{det} ^{1/n}(\Delta uI-D^2 u)=b(x)f(u)\) ${\text{det}}^{1/n}(\mathrm{\Delta }uI-D^{2}u)=b(x)f(u)$ in \({\mathbb {R}}^n\) ${\mathbb{R}}^n$ , which can be regarded as a generalized Keller–Osserman condition. When b is spherically symmetric, we establish the existence of entire large solutions in radial sense. When b is non-spherically symmetric, we obtain the existence of entire bounded solutions using the standard sub- and super-solution method. Finally, the nonexistence results are extended to Hessian quotient equations for a general function b when f are polynomial and exponential functions, respectively.