PreprintCC BY 4.0arXiv
Positive solutions to semipositone problems with mean curvature operator in Minkowski space
Abstract
We establish the existence of a positive solution for a one-dimensional Minkowski-curvature equation with mixed boundary conditions and a sign-changing nonlinearity. A nonnegative linear shift and the Green kernel of the associated linear problem produce an invariant cone. Principal-eigenvalue comparisons and the fixed point index give cone expansion near zero and compression at infinity. A globally defined auxiliary equation and a first-contact argument show that the resulting solution has slope below one and hence solves the original equation. We also derive a directly verifiable asymptotic criterion.
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BibTeX
@misc{xiao2026positive,
title = {{Positive solutions to semipositone problems with mean curvature operator in Minkowski space}},
author = {Ao Xiao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.05711},
eprint = {2608.05711},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.05711},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Positive solutions to semipositone problems with mean curvature operator in Minkowski space
AU - Ao Xiao
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2608.05711
UR - https://arxiv.org/abs/2608.05711
AB - We establish the existence of a positive solution for a one-dimensional Minkowski-curvature equation with mixed boundary conditions and a sign-changing nonlinearity. A nonnegative linear shift and the Green kernel of the associated linear problem produce an invariant cone. Principal-eigenvalue comparisons and the fixed point index give cone expansion near zero and compression at infinity. A globally defined auxiliary equation and a first-contact argument show that the resulting solution has slope below one and hence solves the original equation. We also derive a directly verifiable asymptotic criterion.
ER -