## Forced vibrations of superquadratic Hamiltonian systems.(English)Zbl 0592.70027

The authors consider the Hamiltonian system ($$\circ)$$ $$\dot z=gH'(z)+f(t)$$ where g is the standard skewsymmetric matrix, $$z=z(t)=(p,q): {\mathbb{R}}\to {\mathbb{R}}^{2N}$$, $$H: {\mathbb{R}}^{2N}\to {\mathbb{R}}$$ is a given Hamiltonian and $$f:{\mathbb{R}}\to {\mathbb{R}}^{2N}$$ is an assigned T-periodic function. It is supposed that H satisfies some suitable conditions; in particular the Hamiltonian is required to be superquadratic as $$| z| \to +\infty$$. Under these circumstances it is proved that the system ($$\circ)$$ has infinitely many distinct T- periodic solutions $$\{z_ k\}_{k\in N}$$ and $$\| z_ k\|_ L\to +\infty$$ as $$k\to +\infty$$. This is the main result of the paper. The method to prove it lies in constructing critical topologically stable values for the Lagrangian functional associated with the autonomous system. Then by using Morse theory the authors show that there exist infinitely many critical values for some perturbations of the autonomous functional. In the last part of this lengthy paper the authors prove a more general version of a result obtained by Rabinowitz [one periodic solution, for a system of the type $$\dot z=gH'_ z(t,z)$$, exists; P. H. Rabinowitz, Commun. Pure Appl. Math. 31, 156-184 (1978; Zbl 0358.70014)].
Reviewer: A.Muracchini

### MSC:

 70K40 Forced motions for nonlinear problems in mechanics 70H05 Hamilton’s equations 34C25 Periodic solutions to ordinary differential equations

### Citations:

Zbl 0369.70017; Zbl 0358.70014
Full Text:

### References:

 [1] Adams, R. A.,Sobolev spaces. Academic Press, New York (1975). · Zbl 0314.46030 [2] Amann, H., Multiple periodic solutions of autonomous Hamiltonian systems. InRecent contributions to nonlinear partial differential equations H. Berestycki & H. Brezis editors. Pitman, London (1981). · Zbl 0459.34024 [3] Amann, H. &Zehnder, E., Nontrivial solutions for a class of nonresonance problems and applications.Ann. Scuola Norm. Sup. Pisa (4), 7 (1980), 593–603. · Zbl 0452.47077 [4] Bahri, A.,Thèse de Doctorat d’Etat. Univ. P. et M. Curie, Paris, 1981. [5] Bahri, A. Groupes d’homotopie des ensembles de niveau pour certaines fonctionnelles à gradient Fredholm. To appear. [6] Bahri, A., Résultats topologiques sur une certaine classe de fonctionnelles et applications à l’étude de certains problèmes elliptiques sur-linéaires.J. Funct. Anal. To appear. [7] Bahri, A. &Berestycki, H., A perturbation method in critical point theory and applications.Trans. Amer. Math. Soc., 267 (1981), 1–32. · Zbl 0476.35030 [8] –, Points critiques de perturbations de fonctionnelles paires et applications.C. R. Acad. Sci. Paris. Sér. A, 291 (1980), 189–192. · Zbl 0454.35041 [9] –, Existence d’une infinité de solutions périodiques de certains systèmes hamiltoniens en présence d’un terme de contrainte.C. R. Acad. Sci. Paris Sér. A 292 (1981), 315–318. · Zbl 0471.70019 [10] Bahri, A. & Berestycki, H., Existence of forced oscillations for some nonlinear differential equations.Comm. Pure Appl. Math. To appear. · Zbl 0588.34028 [11] Benci, V., To appear. [12] Benci, V., A geometrical index for the groupsS 1 and some applications to the research of periodic solutions of O.D.E.’s. To appear. [13] Benci, V., On the critical point theory for indefinite functionals in the presence of symmetries.Trans. Amer. Math. Soc., To appear. · Zbl 0504.58014 [14] Benci, V. &Rabinowitz, P. H., Critical point theorems for indefinite functionals.Invent. Math., 52 (1979), 241–273. · Zbl 0465.49006 [15] Brézis, H., Coron, J. M. &Nirenberg, L., Free vibrations for a nonlinear wave equation and a theorem of P. Rabinowitz.Comm. Pure Appl. Math., 33 (1980), 667–689. · Zbl 0484.35057 [16] Clarke, F. H. & Ekeland, I., Nonlinear oscillations and boundary value problems for Hamiltonian systems. To appear. · Zbl 0514.34032 [17] Conner, E. E. &Floyd, F., Fixed point free involutions and equivariant maps.Bull. Amer. Math. Soc., 66 (1960), 416–441. · Zbl 0106.16301 [18] Ekeland, I., Periodic solutions of Hamiltonian equations and a theorem of P. Rabinowitz.J. Differential Equations, 34 (1979), 523–534. · Zbl 0446.70019 [19] Ekeland, I., Forced oscillations de systèmes Hamiltoniens non linéaires, III.Bull. Soc. Math. France. To appear. [20] Ekeland, I. Oscillations de systèmes Hamiltoniens non linéaires, III.Bull. Soc. Math. France. To appear. [21] Fadell, E., Husseini, S. & Rabinowitz, P. H., OnS 1 versions, of the Borsuk-Ulam Theorem. To appear. · Zbl 0506.58010 [22] Fadell, E. &Rabinowitz, P. H., Generalized cohomological index theories for Lie group action with an application to bifurcation questions for Hamiltonian systems.Invent. Math., 45 (1978), 134–174. · Zbl 0403.57001 [23] Fučik, S.,Solvability of nonlinear equations and boundary value problems. D. Reidel, Boston (1980). [24] Krasnosel’skii, M. A.,Topological methods in the theory of nonlinear integral equations. MacMillan, New York (1964). [25] Lions, J. L. &Magenes, E.,Problèmes aux limites non homogènes et applications, vol. 1. Dunod, Paris (1968). · Zbl 0165.10801 [26] Marino, A. &Prodi, G., Metodi perturbativi nella teoria di Morse.Boll. Un. Mat. Ital., 11 (1975), 1–32. [27] Nirenberg, L., Comments on nonlinear problems.Proceeding of a conference in Catania (Italy), September 1981. To appear. [28] Rabinowitz, P. H., Free vibrations for a semi-linear wave equation.Comm. Pure Appl. Math., 31 (1978), 31–68. · Zbl 0341.35051 [29] –, Periodic solutions of Hamiltonian systems.Comm. Pure Appl. Math., 31 (1978), 157–184. · Zbl 0369.70017 [30] –, A variational method for finding periodic solutions of differential equations. InNonlinear evolution equations. M. G. Crandall Ed., pp. 225–251. Academic Press, New York (1968). [31] –, On subharmonic solutions of Hamiltonian systems.Comm. Pure Appl. Math., 33 (1980), 609–633. · Zbl 0437.34011 [32] Rabinowitz, P. H., Variational methods for nonlinear eigenvalue problems. InEigenvalues of nonlinear problems. Roma (1974), Ediz. Cremonese. · Zbl 0278.35040 [33] Rabinowitz, P. H., On large norm periodic solutions of some differential equations. To appear. · Zbl 0504.58018 [34] Schwartz, J. T.,Nonlinear functional analysis. Gordon & Breach, New York (1969). · Zbl 0203.14501
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.