Remarks on dynamical systems with weak forces.

*(English)*Zbl 0606.58039In this paper the existence of T-periodic solutions for the forced system of differential equations -ÿ\(=\nabla V(y)+h(t)\) is studied, where h is T-periodic, V is defined in an open, bounded and convex subset \(\Omega \subset {\mathbb{R}}^ N\) and \(V\to -\infty\) as y tends to the boundary of the set \(\Omega\). The existence of at least one solution is proved, requiring convexity of -V near the boundary. Such a solution is found as a critical point of a functional \(\Phi\) defined via the dual action principle [see F. Clarke and I. Ekeland, Commun. Pure Appl. Math. 33, 103-116 (1980; Zbl 0403.70016)] and, for a situation very close to the one discussed here, see A. Ambrosetti and the author [Solutions with minimal period for Hamiltonian systems in a potential well, Ann. Inst. Henri Poincaré, Anal. Non Linéaire (in print)]. The fact that the critical points of \(\Phi\) correspond to T-periodic solutions of class \(W^{2,1}\) is then exploited to prove that such a solution does not touch the boundary of \(\Omega\).

##### MSC:

37G99 | Local and nonlocal bifurcation theory for dynamical systems |

37J40 | Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion |

##### References:

[1] | A. AMBROSETTI & V. COTI ZELATI, Solutions with minimal period for Hamiltonian systems in a potential well, to appear on Ann. I. H. P. ?Analyse non lineare? · Zbl 0623.58013 |

[2] | A. AMBROSETTI & V. COTI ZELATI, Critical points with lack of compactness and singular dynamical systems, preprint Scuola Normale Superiore, Pisa, 1986 · Zbl 0642.58017 |

[3] | F. CLARKE & I. EKELAND, Hamiltonian trajectories having prescribed minimal period, Comm. Pure and Appl. Math.33, 103-116 (1980) · Zbl 0428.70029 · doi:10.1002/cpa.3160330202 |

[4] | A. CAPOZZI, C. GRECO & A. SALVATORE, Lagrangian systems in presence of singularities, preprint Università di Bari, 1985 · Zbl 0664.34054 |

[5] | A. CAPOZZI & A. SALVATORE, Periodic solutions of Hamiltonian systems: the case of the singular potential, Proc. NATO-ASI (Singh ed.), (1986), pp.207-216 · Zbl 0601.58031 |

[6] | W. GORDON, Conservative dynamical systems involving strong forces, Trans. Am. Math. Soc.,204 (1975), pp. 113-135 · Zbl 0276.58005 · doi:10.1090/S0002-9947-1975-0377983-1 |

[7] | W. GORDON, A minimizing property of Keplerian orbits, Am. Journ. of Math.,99 (1977), pp. 961-971 · Zbl 0378.58006 · doi:10.2307/2373993 |

[8] | P. RABINOWITZ, Some critical point theorem and applications to semilinear elliptic partial differential equations, Ann. Sc. Norm. Sup. Pisa2, pp. 215-223 · Zbl 0375.35026 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.