具有不定权的超线性常微分方程的正下调和解
Positive subharmonic solutions to superlinear ODEs with indefinite weight
摘要：
研究了二阶非线性常微分方程\begin{equation*} u'' + q(t) g(u) = 0, \end{equation*}的正下调和解，其中$g(u)$在零点和无穷远处都具有超线性增长，且$q(t)$是一个$T$ -周期变号权.在$\int_{0}^{T} q(t) ~\!dt < 0$的锐均值条件下，结合Mawhin的重合度理论和Poincaré-Birkhoff不动点定理，证明了对任意大整数$k$，存在$k$级的正下调和解.此外，当$q(t)$的负部分足够大时，利用仍然基于重合度理论的拓扑方法，我们得到了对任意整数$k\geq2$，$k$阶正次谐波的存在性.
英文摘要：
We study the positive subharmonic solutions to the second order nonlinear ordinary differential equation \begin{equation*} u'' + q(t) g(u) = 0, \end{equation*} 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 $\int_{0}^{T} q(t) ~\!dt < 0$, 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 $k\geq2$.
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