Positive subharmonic solutions to superlinear ODEs with indefinite weight

We study the positive subharmonic solutions to the second order nonlinear ordinary differential equation u'' + q(t) g(u) = 0, where g(u) has superlinear growth both at zero and at infinity, and q(t) is a T-periodic sign-changing weight. Under the sharp mean value condition ∫ q(t) dt < 0, combining Mawhin's coincidence degree theory with the Poincaré-Birkhoff fixed point theorem, we prove that there exist positive subharmonic solutions of order k for any large integer k. Moreover, when the negative part of q(t) is sufficiently large, using a topological approach still based on coincidence degree theory, we obtain the existence of positive subharmonics of order k for any integer k≥2