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 f(0)(T) q(t) dt < 0, combining Mawhin's coincidence degree theory with the Poincare-Birkhoff fixed point theorem, we prove that there exist positive sub harmonic 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.