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October  2022, 21(10): 3247-3261. doi: 10.3934/cpaa.2022098

Multiplicity of periodic solutions for second-order perturbed Hamiltonian systems with local superquadratic conditions

Yan Liu 1, and  Fei Guo 2,,

1. 

School of Mathematics, Tianjin University, Tianjin, 300354, China
2. 

School of Mathematics, Tianjin University, Tianjin Key Laboratory of Brain-Inspired Intelligence Technology, Tianjin, 300354, China

* Corresponding author: Fei Guo

Received  July 2020 Revised  March 2022 Published  October 2022 Early access  June 2022

Fund Project: The second author is supported by National Natural Science Foundation of China (12171355) and Elite Scholar Program in Tianjin University, P.R.China

Consider the following second-order perturbed Hamiltonian systems
$ \left\{\begin{array}{l}{\ddot{u}(t)+\nabla_u F(t,u)=\nabla_{u}G(t,u),\quad t\in{\bf{R}},} \\ {u(0)-u(T)=\dot{u}(0)-\dot{u}(T)=\bf0,\quad T>0,}\end{array}\right. $
where
$ F(t,u)=-K(t,u)+W(t,u) $
,
$ K $
,
$ W $
 are measurable and
$ T- $
periodic in
$ t $
 for all
$ u\in\bf{R}^N $
, continuously differentiable in
$ u $
 for a.e.
$ t\in[0,T] $
 and even in
$ u $
,
$ G\in C^1\left(\bf{R}\times\bf{R}^N,\bf{R}\right) $
 is also
$ T- $
periodic in
$ t $
, but
$ G $
 maybe has no parity in
$ u $
. Assume that
$ W $
 is local superquadratic and
$ G $
 is subquadratic at the infinity and
$ K $
 satisfies "pinched" condition, the existence of infinitely many weak periodic solutions for above perturbed systems is obtained via the Bolle's perturbation method, which generalizes and improves some previous results.
Keywords: second-order perturbed Hamiltonian systems, Bolle's perturbation method, local superquadratic conditions, periodic solutions, multiplicity.
Mathematics Subject Classification: 34C25, 37J40, 37J45, 58E05.
Citation: Yan Liu, Fei Guo. Multiplicity of periodic solutions for second-order perturbed Hamiltonian systems with local superquadratic conditions. Communications on Pure and Applied Analysis, 2022, 21 (10) : 3247-3261. doi: 10.3934/cpaa.2022098
References:
[1]

A. Bahri and H. Berestycki, A Perturbation Method in Critical Point Theory and Applications, Trans. Amer. Math. Soc., 267 (1981), 1-32.  doi: 10.2307/1998565.

[2]

P. Bolle, On the Bolza problem, J. Differ. Equ., 152 (1999), 274-288.  doi: 10.1006/jdeq.1998.3484.

[3]

P. Bolle, N. Ghoussoub and H. Tehrani, The multiplicity of solutions in non-homogeneous boundary value problems, Manuscripta Math., 101 (2000), 325-350.  doi: 10.1007/s002290050219.

[4]

G. Fei, On periodic solutions of superquadratic Hamiltonian systems, Electron. J. Differ. Equ., 2002 (2002), 225-228. 

[5]

Y. Liu and F. Guo, Multiplicity of periodic solutions for a class of second-order perturbed Hamiltonian systems, J. Math. Anal. Appl., 491 (2020), 1-14.  doi: 10.1016/j.jmaa.2020.124386.

[6]

Y. Long, Multiple solutions of perturbed superquadratic second order Hamiltonian systems, Trans. Amer. Math. Soc., 311 (1989), 749-780.  doi: 10.2307/2001151.

[7]

Y. Long, Index Theory for Symplectic Paths with Applications, Birkhäuser Verlag, Basel, 2002. doi: 10.1007/978-3-0348-8175-3.

[8]

J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2061-7.

[9]

P. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math., 31 (1978), 157-184.  doi: 10.1002/cpa.3160310203.

[10]

P. Rabinowitz, Multiple critical points of perturbed symmetric functionals, Trans. Amer. Math. Soc., 272 (1982), 753-769.  doi: 10.2307/1998726.

[11]

P. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math. 65, American Mathemayical Society, Providence, 1986. doi: 10.1090/cbms/065.

[12]

A. Salvatore, Multiple homoclinic orbits for a class of second order perturbed Hamiltonian systems, Discrete Contin. Dyn. Syst., 2003 (2003), 778-787. 

[13]

A. Salvatore, Multiple solutions for perturbed elliptic equations in unbounded domains, Adv. Nonlinear Stud., 3 (2003), 1-23.  doi: 10.1515/ans-2003-0101.

[14]

M. Schechter, Periodic solutions of second-order nonautonomous dynamical systems, J. Differ. Equ., 223 (2006), 290-302.  doi: 10.1016/j.jde.2005.02.022.

[15]

M. Struwe, Variational methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer-Verlag, New York, 2000. doi: 10.1007/978-3-540-74013-1.

[16]

Z. Tao, S. Yan and S. Wu, Periodic solutions for a class of superquadratic Hamiltonian systems, J. Math. Anal. Appl., 331 (2007), 152-158.  doi: 10.1016/j.jmaa.2006.08.041.

[17]

Z. Wang and J. Zhang, New existence results on periodic solutions of non-autonomous second order Hamiltonian systems, Appl. Math. Lett., 79 (2018), 43-50.  doi: 10.1016/j.aml.2017.11.016.

[18]

Z. Wang, J. Zhang and Z. Zhang, Periodic solutions of second order non-autonomous Hamiltonian systems with local superquadratic potential, Nonlinear Anal., 70 (2009), 3672-3681.  doi: 10.1016/j.na.2008.07.023.

[19]

Y. Yi and C. Tang, Infinitely many periodic solutions of non-autonomous second-order Hamiltonian systems, Proc. Roy. Soc. Edinb. Sect. A, 144 (2014), 205-223.  doi: 10.1017/S0308210512001461.

[20]

L. Zhang, X. Tang and Y. Chen, Infinitely many homoclinic solutions for a class of indefinite perturbed second-order Hamiltonian systems, I, Mediterr. J. Math., 13 (2016), 3673-3690.  doi: 10.1007/s00009-016-0708-6.

[21]

Q. Zhang and C. Liu, Infinitely many periodic solutions for second order Hamiltonian systems, J. Differ. Equ., 251 (2011), 816-833.  doi: 10.1016/j.jde.2011.05.021.

[22]

Q. Zhang and X. Tang, New existence of periodic solutions for second order non-autonomous Hamiltonian systems, J. Math. Anal. Appl., 369 (2010), 357-367.  doi: 10.1016/j.jmaa.2010.03.033.

show all references

References:
[1]

A. Bahri and H. Berestycki, A Perturbation Method in Critical Point Theory and Applications, Trans. Amer. Math. Soc., 267 (1981), 1-32.  doi: 10.2307/1998565.

[2]

P. Bolle, On the Bolza problem, J. Differ. Equ., 152 (1999), 274-288.  doi: 10.1006/jdeq.1998.3484.

[3]

P. Bolle, N. Ghoussoub and H. Tehrani, The multiplicity of solutions in non-homogeneous boundary value problems, Manuscripta Math., 101 (2000), 325-350.  doi: 10.1007/s002290050219.

[4]

G. Fei, On periodic solutions of superquadratic Hamiltonian systems, Electron. J. Differ. Equ., 2002 (2002), 225-228. 

[5]

Y. Liu and F. Guo, Multiplicity of periodic solutions for a class of second-order perturbed Hamiltonian systems, J. Math. Anal. Appl., 491 (2020), 1-14.  doi: 10.1016/j.jmaa.2020.124386.

[6]

Y. Long, Multiple solutions of perturbed superquadratic second order Hamiltonian systems, Trans. Amer. Math. Soc., 311 (1989), 749-780.  doi: 10.2307/2001151.

[7]

Y. Long, Index Theory for Symplectic Paths with Applications, Birkhäuser Verlag, Basel, 2002. doi: 10.1007/978-3-0348-8175-3.

[8]

J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2061-7.

[9]

P. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math., 31 (1978), 157-184.  doi: 10.1002/cpa.3160310203.

[10]

P. Rabinowitz, Multiple critical points of perturbed symmetric functionals, Trans. Amer. Math. Soc., 272 (1982), 753-769.  doi: 10.2307/1998726.

[11]

P. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math. 65, American Mathemayical Society, Providence, 1986. doi: 10.1090/cbms/065.

[12]

A. Salvatore, Multiple homoclinic orbits for a class of second order perturbed Hamiltonian systems, Discrete Contin. Dyn. Syst., 2003 (2003), 778-787. 

[13]

A. Salvatore, Multiple solutions for perturbed elliptic equations in unbounded domains, Adv. Nonlinear Stud., 3 (2003), 1-23.  doi: 10.1515/ans-2003-0101.

[14]

M. Schechter, Periodic solutions of second-order nonautonomous dynamical systems, J. Differ. Equ., 223 (2006), 290-302.  doi: 10.1016/j.jde.2005.02.022.

[15]

M. Struwe, Variational methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer-Verlag, New York, 2000. doi: 10.1007/978-3-540-74013-1.

[16]

Z. Tao, S. Yan and S. Wu, Periodic solutions for a class of superquadratic Hamiltonian systems, J. Math. Anal. Appl., 331 (2007), 152-158.  doi: 10.1016/j.jmaa.2006.08.041.

[17]

Z. Wang and J. Zhang, New existence results on periodic solutions of non-autonomous second order Hamiltonian systems, Appl. Math. Lett., 79 (2018), 43-50.  doi: 10.1016/j.aml.2017.11.016.

[18]

Z. Wang, J. Zhang and Z. Zhang, Periodic solutions of second order non-autonomous Hamiltonian systems with local superquadratic potential, Nonlinear Anal., 70 (2009), 3672-3681.  doi: 10.1016/j.na.2008.07.023.

[19]

Y. Yi and C. Tang, Infinitely many periodic solutions of non-autonomous second-order Hamiltonian systems, Proc. Roy. Soc. Edinb. Sect. A, 144 (2014), 205-223.  doi: 10.1017/S0308210512001461.

[20]

L. Zhang, X. Tang and Y. Chen, Infinitely many homoclinic solutions for a class of indefinite perturbed second-order Hamiltonian systems, I, Mediterr. J. Math., 13 (2016), 3673-3690.  doi: 10.1007/s00009-016-0708-6.

[21]

Q. Zhang and C. Liu, Infinitely many periodic solutions for second order Hamiltonian systems, J. Differ. Equ., 251 (2011), 816-833.  doi: 10.1016/j.jde.2011.05.021.

[22]

Q. Zhang and X. Tang, New existence of periodic solutions for second order non-autonomous Hamiltonian systems, J. Math. Anal. Appl., 369 (2010), 357-367.  doi: 10.1016/j.jmaa.2010.03.033.

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