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Subharmonic solutions of conservative systems with nonconvex potentials. (English) Zbl 0752.34027

The authors consider the system of second order differential equations (1) \(u''+\nabla G(u)=e(t)\). Here, \(G: \mathbb{R}^ n\to\mathbb{R}'\) is a continuously differentiable function with gradient \(\nabla G\) and \(e: \mathbb{R}'\to\mathbb{R}^ n\) is a continuous periodic function having a minimal period \(T>0\). They prove the existence of subharmonic solutions without assuming the convexity of \(G\) by simply making some careful estimates on the critical levels of the functionals associated to the problem. The main result concerning the system is the following theorem:
Assume that the range of \(\nabla G(u)\) for \(u\in\mathbb{R}^ n\) is bounded, and moreover \[ \lim_{\| u\|\to\infty}\langle\nabla G(u)-\bar e,u\rangle=\infty,\quad\left(\bar e={1\over T}\int^ T_ 0 e(t)dt\right). \] Then system (1), besides having at least one \(T\)- periodic solution, also has periodic solutions with minimal period \(kT\), for any sufficiently large prime number \(k\).

MSC:

34C25 Periodic solutions to ordinary differential equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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