Rich dynamics in planar systems with heterogeneous nonnegative weights

This paper studies the global structure of set nodal solutions a generalized Sturm–Liouville boundary value problem associated to quasilinear equation$ -(\phi(u'))' = \lambda u + a(t)g(u), \quad \lambda\in {\mathbb R}, $where $ a(t) is non-negative with some positive humps separated away by intervals degeneracy where a\equiv 0 $. When \phi(s) s this equation includes prototype classical model going back Moore and Nehari [35], 1959. first general case when {\lambda}\in\mathbb{R} has been addressed a\gneq The semilinear a\lneq recently treated López-Gómez Rabinowitz [28,29,30].