Lorentz--Minkowski平均曲率方程的径向结点解：坍塌与饱和

中文摘要 AI 辅助

我们考虑径向设定下Lorentz--Minkowski平均曲率算子的Dirichlet问题，即 $$ -\text{div}\frac{\nabla u}{\sqrt{1-|\nabla u|^2}} = \lambda u + \mu g(u) \text{ 于 } \mathcal B_R, \u2003 u=0 \text{ 于 } \partial\mathcal B_R, $$ 定义在原点为中心、半径为$R$的球$\mathcal B_R \subset \mathbb{R}^N$上，其中$\lambda, \mu \geq 0$为参数，非线性项$g$在零点满足适当的增长假设。我们首先证明当$\mu \to +\infty$时，具有给定零点数的解的渐近行为的一个先验二分法：要么它们一致收敛于$0$，要么它们收敛于一个分段仿射极限函数$u_\infty$，且$|u_\infty'|=1$几乎处处成立，并具有相同的结点性质。然后我们证明，当零点数被适当给定时，这两种情形实际上都能实现。更具体地说，基于[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)]中的多重性结果，我们细化了其中的打靶构造，以获得一族收敛于零的小解，并通过向后打靶论证，获得一族收敛于饱和分段仿射轮廓的大解。

英文摘要

We consider the Dirichlet problem for the Lorentz--Minkowski mean curvature operator in a radial setting, $$ -\operatorname{div}\left(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right) = λu + μg(u) \quad \text{in } \mathcal B_R, \qquad u = 0 \quad \text{on } \partial\mathcal B_R, $$ on a ball of radius $R$ centered at the origin $\mathcal B_R \subset \mathbb{R}^N$, where $λ, μ\geq 0$ are parameters, and the nonlinearity $g$ satisfies suitable growth assumptions at zero. We first prove an a priori dichotomy for the asymptotic behaviour, as $μ\to +\infty$, 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 $|u_\infty'|=1$ 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.

发表机构

• Università di Torino（都灵大学）

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