Positive subharmonic solutions to nonlinear ODEs with indefinite weight

2016/05/09 by Boscaggin, Alberto, Feltrin, Guglielmo
#34B18 #34C25 (Primary) #37J10 #47H11 (Secondary) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1605.02500

Abstract

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 ∫₀^T 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).

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