Collapse and saturation for radial nodal solutions of a Lorentz--Minkowski mean curvature equation

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 $