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$ -(\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 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 [35], 1959. This is the first paper where the general case when $ {\lambda}\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 [28,29,30].