Sharp conditions for fully nonlinear equations with $(n-1)$-Hessian matrix: Part I. Monge--Ampère case

We study positive $(n-1)$-convex entire subsolutions of the normalized real $(n-1)$ Monge–Ampère equation ${\det }^{\frac{1}{n}}(\mathrm{\Delta }uI-D^{2}u)=f(u)+g(u)\mathrm{\mid }Du{\mathrm{\mid }}^q$ in ${\mathbb{R}}^n$. Under two sets of structural assumptions on the continuous, non-decreasing coefficients $f$ and $g$, we establish necessary and sufficient conditions for the existence of entire subsolutions when $q<2-\frac{1}{n}$. Our results cover the pure-gradient degenerate case $f\equiv 0$ with $0<q<1$, the gradient-singular case $q<0$, and the case where $g$ changes sign. For $q=2-\frac{1}{n}$, we recover the Keller–Osserman condition for the gradient-independent equation, prove nonexistence when $g$ is eventually positive, and obtain an exponentially weighted condition when $g$ has a nonpositive limit at infinity. This is the first of two companion papers and treats the determinant case.