Multiple solutions to Neumann problems with indefinite weight and bounded nonlinearities

We study the Neumann boundary value problem for the second order ODE \begin{equation}\label{eqabs} u'' + (a^+(t)-\mu a^-(t))g(u) = 0, \qquad t \in [0,T], \end{equation} where $g \in \mathcal{C}^1(\mathbb{R})$ is a bounded function of constant sign, $a^+,a^-: [0,T] \to \mathbb{R}^+$ are the positive/negative part of a sign-changing weight $a(t)$ and $\mu > 0$ is a real parameter. Depending on the sign of $g'(u)$ at infinity, we find existence/multiplicity of solutions for $\mu$ in a ``small'' interval near the value $$ \mu_c = \frac{\int_0^T a^+(t) \, dt}{\int_0^T a^-(t) \, dt}\,. $$ The proof exploits a change of variables, transforming the sign-indefinite equation \eqref{eqabs} into a forced perturbation of an autonomous planar system, and a shooting argument. Nonexistence results for $\mu \to 0^+$ and $\mu \to +\infty$ are given, as well.