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September  2016, 15(5): 1625-1642. doi: 10.3934/cpaa.2016005

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

Xiao-Fei Zhang 1, and  Fei Guo 1,

1. 

Department of Mathematics, School of Science, Tianjin University, Tianjin 300072, China, China

Received  March 2015 Revised  August 2015 Published  July 2016

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: Subharmonic solutions, Hamiltonian systems., super-quadratic condition, multiplicity, periodic solutions.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Xiao-Fei Zhang, Fei Guo. Multiplicity of subharmonic solutions and periodic solutions of a particular type of super-quadratic Hamiltonian systems. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1625-1642. doi: 10.3934/cpaa.2016005
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.

show all references

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.

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