# American Institute of Mathematical Sciences

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

 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
 $W$
 $G$
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.
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.

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##### 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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