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Positive subharmonic solutions to superlinear ODEs with indefinite weight

Work partially supported by the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). Progetto di Ricerca 2016: "Problemi differenziali non lineari: esistenza, molteplicità e proprietà qualitative delle soluzioni". It is also partially supported by the project "Existence and asymptotic behavior of solutions to systems of semilinear elliptic partial differential equations" (T.1110.14) of the Fonds de la Recherche Fondamentale Collective, Belgium.
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  • 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 $\int_{0}^{T}{q\left( t \right)dt<0}$ , 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 $k≥2$ .

    Mathematics Subject Classification: Primary: 34C25; Secondary: 34B18, 37J10, 47H11.


    \begin{equation} \\ \end{equation}
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  • Figure 1.  The figure shows an example of multiple positive solutions for the $T$-periodic boundary value problem associated with $(\mathscr{E}_{\mu})$. For this numerical simulation we have chosen $a(t) = \sin(6\pi t)$ for $t\in\mathopen{[}0, 1\mathclose{]}$, $\mu = 10$ and $g(s) = \max\{0,400\, s\arctan|s|\}$. Notice that the weight function $a(t)$ has $3$ positive humps. We show the graphs of the $7$ positive $T$-periodic solutions of $(\mathscr{E}_{\mu})$. We stress that $g(s)/s\not\to+\infty$ as $s\to+\infty$ (contrary to what is assumed in $(g_{s})$), indeed Theorem 2.2 is also valid assuming only that $g(s)/s$ is sufficiently large as $s\to+\infty$, as observed in Remark 2.1

    Figure 2.  Using a numerical simulation we have studied the $2$-periodic solutions of equation $(\mathscr{E}_{\mu})$ where $a(t) = \sin(t)$, $T=2\pi$, $\mu = 6$ and $g(s) = 100(s^{2}+s^{3})$. In the upper part, the figure shows the graph of the weight $q(t)=a_{\mu}(t)$; while in the lower part, it shows the graphs of three $2T$-periodic positive solutions, in accordance with Theorem 4.1. The three solutions are in correspondence with the three strings $(1, 0)$, $(0, 1)$ and $(1, 1)$, respectively, as stated in Theorem 4.1. The first two solutions are subharmonic solutions of order $2$ and the third one is a $1$-periodic solution. As subharmonic solutions of order $2$, we consider only the first one, since the second one is a translation by $1$ of the first solution

    Figure 3.  The figure shows the aperiodic necklaces made by arranging $k$ beads whose color is chosen from a list of 2 colors, when $k\in\{2, 3, 4\}$. The aperiodic necklaces depicted in the figure correspond to the following Lyndon words on the alphabet $\{a, b\}$: $ab$ for $k=2$, $abb$ and $aab$ for $k=3$, $abbb$, $aabb$ and $aaab$ for $k=4$ (which are the minimal elements in the class of equivalence in the lexicographic ordering). For instance, for $k = 4$, note that the string $abab$ is not acceptable as it represents a sequence of period $2$ and the string $bbaa$ is already counted as $aabb$

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