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arXiv · 2610.15693

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

Jingwen Ji·Feida Jiang·Mengni Li
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Abstract

We study positive $(n-1)$-convex entire subsolutions of the normalized real $(n-1)$ Monge--Ampère equation $\operatorname{det}^{\frac1n}\bigl(ΔuI-D^2u\bigr)=f(u)+g(u)|Du|^q$ 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 $q<2-\frac1n$. Our results cover the pure-gradient degenerate case $f\equiv0$ with $0<q<1$, the gradient-singular case $q<0$, and the case where $g$ changes sign. For $q=2-\frac1n$, 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.

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Jingwen Ji, Feida Jiang, Mengni Li. 2026-10-11. Sharp conditions for fully nonlinear equations with $(n-1)$-Hessian matrix: Part I. Monge--Ampère case. https://arxiv.org/abs/2610.15693

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