Positive solutions with a complex behavior for superlinear indefinite ODEs on the real line

We show the existence of infinitely many positive solutions, defined on the real line, for the nonlinear scalar ODEu¨+(a+(t)−μa−(t))u3=0, where a   is a periodic, sign-changing function, and the parameter μ>0μ>0 is large. Such solutions are characterized by the fact of being either small or large in each interval of positivity of a. In this way, we find periodic solutions, having minimal period arbitrarily large, and bounded non-periodic solutions, exhibiting a complex behavior. The proof is variational, exploiting suitable natural constraints of Nehari type.