Rich dynamics in planar systems with heterogeneous nonnegative weights

Abstract

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), λ ∈ R, where a(t) is non-negative with some positive humps separated away by intervals of degeneracy where a ≡ 0. When φ(s) = s this equation includes a generalized prototype of a classical model going back to Moore and Nehari [35], 1959. This is the first paper where the general case when λ ∈ R has been addressed when a ≥ 0. The semilinear case with a ≤ 0 has been recently treated by López-Gómez and Rabinowitz [28, 29, 30].