Positive subharmonic solutions to nonlinear ODEs with indefinite weight

Abstract

We prove that the superlinear indefinite equation \begin{equation*} u" + a(t)u^{p} = 0, \end{equation*} where $p>1$ and $a(t)$ is a $T$-periodic sign-changing function satisfying the (sharp) mean value condition ${\int }_0^{T}a(t)dt<0$, 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).

26 pages