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Remarks on dynamical systems with weak forces. (English) Zbl 0606.58039
In 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
##### Keywords:
periodic solution; singular potential
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##### 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
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