Positive subharmonic solutions to nonlinear ODEs with indefinite weight

We prove that the superlinear indefinite equation u'' + a(t)u^p = 0, where p > 1 and a(t) is a T-periodic sign-changing function satisfying the (sharp) mean value condition ∫a(t)dt<0, has positive subharmonic solutions of order k for any large integer k, thus providing a further contribution to a problem raised by G. J. Butler in its pioneering paper [JDE, 1976]. The proof, which applies to a larger class of indefinite equations, combines coincidence degree theory (yielding a positive harmonic solution) with the Poincaré-Birkhoff fixed point theorem (giving subharmonic solutions oscillating around it).