On second order, periodic, symmetric, differential systems having subharmonics of all sufficiently large orders
This article deals with the second order differential system (1) \(u'' (t) + g(u (t)) = e(t)\), where \(u : \mathbb{R} \to \mathbb{R}^N\), \(N \geq 1\), \(g = \nabla G\) with \(G \in C^1 (\mathbb{R}^N, \mathbb{R})\), and \(e : \mathbb{R} \to \mathbb{R}^N\) is a function which is continuous and periodic in \(t\), with minimal period \(2 \pi\). A periodic solution \(u\) of equation (1) is called a subharmonic of order \(k\) if \(2k \pi\) is the least positive period of \(u\). In preceding papers, some authors proved the existence of infinitely many subharmonics under suitable assumptions of subquadratic growth at infinity for the function \(G\). Some previous results of this kind (see, e.g. [\textit{A. Fonda} and \textit{A. C. Lazer}, Proc. Am. Math. Soc. 115, 183-190 (1992; Zbl 0752.34027)], or [\textit{A. Fonda} and \textit{M. Ramos}, J. Differential Equations 109, 354-372 (1994; Zbl 0798.34048)], ensured the existence of subharmonics of order \(p\), for each sufficiently large prime number \(p\). The same conclusion is achieved in Theorem 2.1 of this article, with respect to the existence of a sequence of large amplitude odd subharmonics, under the assumptions: \(G \) even, \(e (\cdot)\) odd, \(G (\xi) \to \infty\) and \(G(\xi)/ |\xi |^2 \to 0\) as \(|\xi |\to \infty\). In the main result of the paper (Theorem 3.1), the authors address their interest to the search of subharmonics of order \(k\), for each sufficiently large integer \(k\). This conclusion, which is usually much harder to prove, is obtained here under the supplementary condition \[ \exists \theta \in (0,2),\;\exists b : \theta G (\xi) \leq g (\xi) \cdot \xi + b,\;\forall \xi \in \mathbb{R}^N. \] The proof makes use of a variational approach, providing, for each \(k \geq 1\), a \(2k \pi\)-periodic solution \(u_k\) obtained as a minimum of the functional associated to (1) in a space of odd \(2k \pi\)-periodic functions. It is also proved that \(|u_k |_\infty \to \infty\) as \(k \to \infty\). The key point in order to guarantee the minimality of the period for \(u_k\), follows from some energy estimates which imply that \(r^2_k = |u_k' |^2_{L^2 (-k \pi, k \pi)}/k \to + \infty\) as \(k \to \infty\) and that there is a bound \(M\) (independent of \(k)\), such that \(r_k < M\) if the minimal period of \(u_k\) is less than \(2k \pi\) (Lemma 3.2 and Lemma 3.3, respectively). It can be remarked that in the case of the scalar Duffing's equations \((N = 1)\), {it F. Zandin} and the reviewer [Nonlinear Anal., Theory Methods Appl. 20, 509-532 (1993; Zbl 0778.34027)] already proved the existence of large amplitude subharmonics of order \(k\), for each sufficiently large integer \(k\), as well as obtained information on their nodal properties, using the one-sided subquadratic growth condition \(G(\xi)/ \xi^2 \to 0\) as \(\xi \to + \infty\).
scientific article; zbMATH DE number 885508