## Solutions with minimal period for Hamiltonian systems in a potential well.(English)Zbl 0623.58013

Let $$U\in C^ 2(\Omega)$$, where $$\Omega$$ is a bounded set in $${\mathbb{R}}^ N$$. Suppose that U(x) tends to $$+\infty$$ as x tends to $$\partial \Omega$$. Our main results concern the existence of periodic solutions of - ẍ$$=U'(x)$$ having a prescribed number T as minimal period. The results are also generalized to first order Hamiltonian systems.

### MSC:

 37G99 Local and nonlocal bifurcation theory for dynamical systems 49J35 Existence of solutions for minimax problems

### Keywords:

potential well; periodic solutions; Hamiltonian systems
Full Text:

### References:

 [1] Ambrosetti, A., Nonlinear oscillatiations with minimal period, Proceed. Symp. Pure Math., Vol. 44, 29-35, (1985) [2] Ambrosetfi, A.; Mancini, G., Solutions of minimal period for a class of convex Hamiltonian systems, Math. Ann., Vol. 255, 405-421, (1981) · Zbl 0466.70022 [3] Ambrosetti, A.; Rabinowitz, P., Dual variational methods in critical point theory and applications, J. Funct. Anal., Vol. 14, 349-381, (1973) · Zbl 0273.49063 [4] Aubin, J. P.; Ekeland, I., Applied nonlinear analysis, (1984), Wiley New York [5] Benci, V., Normal modes of a Lagrangian system constrained in a potential well, Ann. I.H.P. “Analyse non lineare”, Vol. 1, 379-400, (1984) · Zbl 0561.58006 [6] Clarke, F., Periodic solutions of Hamiltonian inclusions, J. Diff. Eq., Vol. 40, 1-6, (1981) · Zbl 0461.34030 [7] Clarke, F., Optimization and nonsmooth analysis, (1983), Wiley New York · Zbl 0582.49001 [8] Clarke, F.; Ekeland, I., Hamiltonian trajectories having prescribed minimal period, Comm. Pure and Appl. Math., Vol. 33, 103-116, (1980) · Zbl 0403.70016 [9] Ekeland, I., Periodic solutions to Hamiltonian equations and a theorem od P. Rabinowitz, J. Diff. Eq., Vol. 34, 523-534, (1979) · Zbl 0446.70019 [10] Ekeland, I., Une théorie de Morse pour LES systèmes hamiltoniens convexes, Ann. I.H.P. “Analyse non lineare”, Vol. 1, 19-78, (1984) · Zbl 0537.58018 [11] Ekeland, I.; Hofer, H., Periodic solutions with prescribed period for convex autonomous Hamiltonian systems, Inv. Math., 81, 155-188, (1985) · Zbl 0594.58035 [12] Girardi, M.; Matzeu, M., Periodic solutions of convex Hamiltonian systems with a quadratic growth at the origin and superquadratic at infmity, (1985), preprint, Univ. degli Studi di Roma Roma [13] Girardi, M.; Matzeu, M., Some results on solutions of minimal period to Hamiltonian systems, (Ambrosetti, A., Nonlinear Oscillations for Conservative Systems, (1985), Pitagora Bologna), 27-35 [14] Kufner, A.; John, O.; Fucik, S., Function spaces, (1977), Academia Prague [15] Rabinowitz, P., Periodic solutions of Hamiltonian systems, Comm. Pure and Appl. Math., Vol. 31, 157-184, (1978) · Zbl 0358.70014 [16] Rabinowitz, P., Periodic solutions of Hamiltonian systems: a survey, S.I.A.M. J. Math. Anal., Vol. 13, 343-352, (1982) · Zbl 0521.58028 [17] Benedetto, J. J., Real variable and integration, (1976), B. G. Teubner Stuttgart · Zbl 0336.26001
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.