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, $$-\mathrm{div}\left(\frac{\nabla u}{\sqrt{1-|\nabla u{|}^2}}\right)=\lambda u+\mu g(u)\text{in }{\mathcal{B}}_R,u=0\text{on }\partial {\mathcal{B}}_R,$$ on a ball of radius $R$ centered at the origin ${\mathcal{B}}_R\subset {\mathbb{R}}^N$, where $\lambda ,\mu \ge 0$ are parameters, and the nonlinearity $g$ satisfies suitable growth assumptions at zero. We first prove an a priori dichotomy for the asymptotic behaviour, as $\mu \to +\infty$, of solutions with a prescribed number of zeros: either they converge uniformly to $0$, or they converge to a piecewise affine limit function $u_{\infty }$, with $|u_{\infty }^{\prime }|=1$ 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.