Multiplicity of subharmonic solutions and periodic solutions of a particular type of super-quadratic Hamiltonian systems

Xiao-Fei  Zhang; Fei  Guo

doi:10.3934/cpaa.2016005

Article Contents

2016, Volume 15, Issue 5: 1625-1642. Doi: 10.3934/cpaa.2016005

This issue Previous Article Decay of the compressible viscoelastic flows Next Article On the trace regularity results of Musielak-Orlicz-Sobolev spaces in a bounded domain

Multiplicity of subharmonic solutions and periodic solutions of a particular type of super-quadratic Hamiltonian systems

Xiao-Fei Zhang¹ and
Fei Guo^1,

1.
Department of Mathematics, School of Science, Tianjin University, Tianjin 300072

Received: February 28, 2015

Revised: July 31, 2015

Published: August 2016

Abstract Related Papers Cited by

Abstract

This paper considers the Hamiltonian systems with new generalized super-quadratic conditions. Using the variational principle and critical point theory, we obtain infinitely many distinct subharmonic solutions and an unbounded sequence of periodic solutions.

Keywords:

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

Citation:

Related Papers

Cited by

References

[1]	T. Q. An and Z. Q. Wang, Periodic solutions of Hamiltonian systems with anisotropic growth, Comm. Pure Appl. Ana., 9 (2010), 1069-1082.doi: 10.3934/cpaa.2010.9.1069.
[2]	S. L. Chen and C. H. Tang, Periodic and subharmonic solutions of a class of superquadratic Hamiltonian systems, J. Math. Anal. Appl., 297 (2004), 267-284.doi: 10.1016/j.jmaa.2004.05.006.
[3]	G. H. Fei, On periodic solutions of superquadratic Hamiltonian systems, Electro. J. Differential Equations, 8 (2002), 1-12.
[4]	P. L. Felmer, Periodic solutions of "superquadratic'' Hamiltonian systems, J. Differential Equations, 102 (1993), 188-207.doi: 10.1006/jdeq.1993.1027.
[5]	P. L. Felmer and Z. Q. Wang, Multiplicity for symmetric indefinite functionals: applications to Hamiltonian systems and elliptic systems, Topol. Methods Nonlinear Anal., 12 (1998), 207-226.
[6]	C. Li and C. G. Liu, Brake subharmonic solutions of first order Hamiltonian systems, Sci. China Math., 53 (2010), 2719-2732.doi: 10.1007/s11425-010-4105-5.
[7]	C. G. Liu, Subharmonic solutions of Hamiltonian systems, Nonlinear Anal., 42 (2000), 185-198.doi: 10.1016/S0362-546X(98)00339-3.
[8]	Y. Long, Periodic solutions of perturbed super-quadratic Hamiltonian systems, Ann. Scuola Norm. Sup. Pisa. Serie IV. Vol. XVII. Fasc., 1 (1990), 35-77.
[9]	Y. Long, Periodic solutions of Hamiltonian systems with bounded forcing terms, Math. Z., 203 (1990), 453-467.doi: 10.1007/BF02570749.
[10]	J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer-Verlag, New York, 1989.doi: 10.1007/978-1-4757-2061-7.
[11]	P. H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math., 31 (1978), 157-184.
[12]	P. H. Rabinowitz, On subhamonic solutions of Hamiltonian systems, Comm. Pure Appl. Math., 33 (1980), 609-633.doi: 10.1002/cpa.3160330504.
[13]	P. H. Rabinowitz, Periodic solutions of large norm of Hamiltonian systems, J. Differential Equations, 50 (1986), 33-48.doi: 10.1016/0022-0396(83)90083-9.
[14]	P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math. 65, American Mathematical Society, Providence, 1986.doi: 10.1090/cbms/065.
[15]	C. H. Tang, Existence and multiplicity of periodic solutions for nonautonomous second order systems, Nonlinear Anal., 32 (1998), 299-304.doi: 10.1016/S0362-546X(97)00493-8.
[16]	X. J. Xu, Periodic solutions for non-autonomous Hamiltonian systems possessing super-quardratic potentials, Nonlinear Anal., 51 (2002), 941-955.doi: 10.1016/S0362-546X(01)00870-7.
[17]	Q. Y. Zhang and C. G. Liu, Infinitely many periodic solutions for second order Hamiltonian systems, J. Differential Equations, 251 (2011), 816-833.doi: 10.1016/j.jde.2011.05.021.
[18]	X. F. Zhang and F. Guo, Existence of periodic solutions of a particular type of super-quadratic Hamiltonian systems, J. Math. Anal. Appl., 421 (2015), 1587-1602.doi: 10.1016/j.jmaa.2014.08.006.
[19]	W. Zou and S. Li, Infinitely many solutions for Hamiltonian systems, J. Differential Equations, 186 (2002), 141-164.doi: 10.1016/S0022-0396(02)00005-0.