Variational characterization of nodal solution families

Investigate whether a variational interpretation exists for the radial nodal solution families with at least two nodal regions, including whether the large nodal solutions admit a minimizing characterization and whether the small nodal solutions can be detected through suitable higher min–max constructions.

Background

The paper constructs, for each prescribed nodal count, two families of radial solutions of the Lorentz–Minkowski mean curvature equation as the parameter $\mu$ tends to infinity. The small solutions converge uniformly to zero, while the large solutions converge to saturated piecewise-affine profiles whose slopes have absolute value one. For positive solutions, corresponding variational interpretations are already known: the min–max solution exhibits collapse to zero and the global minimizer exhibits saturation toward the distance function.

For nodal solutions with $\ell\geq2$, the paper establishes the asymptotic existence and behavior by shooting methods rather than by variational arguments. It leaves unresolved whether the two nodal families can similarly be identified through variational principles, specifically through minimization for the large solutions and higher min–max constructions for the small solutions.