Collapse and saturation for radial nodal solutions of a Lorentz--Minkowski mean curvature equation
Abstract
We consider the Dirichlet problem for the Lorentz--Minkowski mean curvature operator in a radial setting, −div((∇ u)/(√(1−|∇ u|2))) = λu + μg(u) in ℬR, u = 0 on ∂ℬR, on a ball of radius R centered at the origin ℬR ⊂ ℝN, where λ, μ≥ 0 are parameters, and the nonlinearity g satisfies suitable growth assumptions at zero. We first prove an a priori dichotomy for the asymptotic behaviour, as μ→ +∞, of solutions with a prescribed number of zeros: either they converge uniformly to $0$, or they converge to a piecewise affine limit function u∞, with |u∞'|=1 a.e., having the same nodal properties. We then show that, when the number of zeros is appropriately prescribed, both alternatives are actually realized. More precisely, building on the multiplicity result in [A. Boscaggin, F. Colasuonno, and R. Ziegele, \emph{Positive and nodal solutions for the Minkowski mean curvature equation: multiplicity and asymptotics}, arXiv:2607.15956 (2026)], we refine the shooting construction therein to obtain a family of small solutions converging to zero and, through a backward shooting argument, a family of large solutions converging to a saturated piecewise affine profile.
Related
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BibTeX
@misc{boscaggin2026collapse,
title = {{Collapse and saturation for radial nodal solutions of a Lorentz--Minkowski mean curvature equation}},
author = {Alberto Boscaggin and Francesca Colasuonno and Ricardo Ziegele},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10166},
eprint = {2610.10166},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10166},
}
RIS
TY - UNPB
TI - Collapse and saturation for radial nodal solutions of a Lorentz--Minkowski mean curvature equation
AU - Alberto Boscaggin
AU - Francesca Colasuonno
AU - Ricardo Ziegele
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10166
UR - https://arxiv.org/abs/2610.10166
AB - We consider the Dirichlet problem for the Lorentz--Minkowski mean curvature operator in a radial setting, −div((∇ u)/(√(1−|∇ u|^2))) = λu + μg(u) in ℬ_R, u = 0 on ∂ℬ_R, on a ball of radius R centered at the origin ℬ_R ⊂ ℝ^N, where λ, μ≥ 0 are parameters, and the nonlinearity g satisfies suitable growth assumptions at zero. We first prove an a priori dichotomy for the asymptotic behaviour, as μ→ +∞, of solutions with a prescribed number of zeros: either they converge uniformly to $0$, or they converge to a piecewise affine limit function u_∞, with |u_∞'|=1 a.e., having the same nodal properties. We then show that, when the number of zeros is appropriately prescribed, both alternatives are actually realized. More precisely, building on the multiplicity result in [A. Boscaggin, F. Colasuonno, and R. Ziegele, \emph{Positive and nodal solutions for the Minkowski mean curvature equation: multiplicity and asymptotics}, arXiv:2607.15956 (2026)], we refine the shooting construction therein to obtain a family of small solutions converging to zero and, through a backward shooting argument, a family of large solutions converging to a saturated piecewise affine profile.
ER -