具有不定权的非线性常微分方程的正下调和解
Positive subharmonic solutions to nonlinear ODEs with indefinite weight
摘要：
本文证明了超线性不定方程\begin{equation*} u" + a(t)u^{p} = 0, \end{equation*}（其中$p > 1$和$a(t)$是满足（尖）均值条件$\int_{0}^{T} a(t)~\!dt < 0$的$T$ -周期变号函数）对任意大整数$k$都有$k$阶的正下调和解，从而对G. J. Butler在其开创性论文（JDE，1976）中。证明，适用于一个更大的一类不定方程，结合重合度理论（产生一个正调和解）与庞加莱-伯克霍夫不动点定理（给下调和解振荡周围）。
英文摘要：
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).
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