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 $-(ϕ(u'))'= λu + a(t)g(u), \quad λ\in {\mathbb R},$ where $a(t)$ is non-negative with some positive humps separated away by intervals of degeneracy where $a\equiv 0$. When $ϕ(s)=s$ 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 $λ\in\mathbb{R}$ has been addressed when $a\gneq 0$. The semilinear case with $a\lneq 0$ has been recently treated by López-Gómez and Rabinowitz.

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