Rich dynamics in planar systems with heterogeneous nonnegative weights

This paper studies the global structure of the set of nodal solutions of a generalized Sturm–Liouville boundary value problem associated to the quasilinear equation $$-(\phi (u^{\mathrm{\prime }}){)}^{\mathrm{\prime }}=\lambda u+a(t)g(u),\lambda \in \mathbb{R},$$ where $a(t)$ is non-negative with some positive humps separated away by intervals of degeneracy where $a\equiv 0$. When $\phi (s)=s$ this equation includes a generalized prototype of a classical model going back to Moore and Nehari, 1959. This is the first paper where the general case when $\lambda \in \mathbb{R}$ has been addressed when $a⪈0$. The semilinear case with $a⪇0$ has been recently treated by López-Gómez and Rabinowitz.