Sharp conditions for fully nonlinear equations with (n−1)-Hessian matrix: Part I. Monge--Ampère case
Abstract
We study positive (n−1)-convex entire subsolutions of the normalized real (n−1) Monge--Ampère equation det1/n(ΔuI−D2u)=f(u)+g(u)|Du|q in ℝn. 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<2−1/n. Our results cover the pure-gradient degenerate case f≡0 with 0<q<1, the gradient-singular case q<0, and the case where g changes sign. For q=2−1/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.
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BibTeX
@misc{ji2026sharp,
title = {{Sharp conditions for fully nonlinear equations with $(n-1)$-Hessian matrix: Part I. Monge--Ampère case}},
author = {Jingwen Ji and Feida Jiang and Mengni Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15693},
eprint = {2610.15693},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15693},
}
RIS
TY - UNPB
TI - Sharp conditions for fully nonlinear equations with (n−1)-Hessian matrix: Part I. Monge--Ampère case
AU - Jingwen Ji
AU - Feida Jiang
AU - Mengni Li
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.15693
UR - https://arxiv.org/abs/2610.15693
AB - We study positive (n−1)-convex entire subsolutions of the normalized real (n−1) Monge--Ampère equation det^1/n(ΔuI−D^2u)=f(u)+g(u)|Du|^q in 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<2−1/n. Our results cover the pure-gradient degenerate case f≡0 with 0<q<1, the gradient-singular case q<0, and the case where g changes sign. For q=2−1/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.
ER -