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April  2019, 24(4): 1617-1625. doi: 10.3934/dcdsb.2018222

## Nonconstant periodic solutions with any fixed energy for singular Hamiltonian systems

 1 School of Mathematics, Sichuan University, Chengdu 610064, China 2 School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China 3 School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan 430073, China

* Corresponding author: Jinlong Wei

Received  January 2018 Revised  March 2018 Published  June 2018

Fund Project: The first author is partially supported by National Science Foundation of China (116712787) and Graduate Student's Research and Innovation Fund of Sichuan University (2018YJSY045). The third author is partially supported by National Science Foundation of China (11501577, 61773401).

In the past years, there were very few works on the existence of nonconstant periodic solutions with fixed energy of singular second-order Hamiltonian systems, and now we attempt to ingeniously use Ekeland's variational principle to prove the existence of nonconstant periodic solutions with any fixed energy for singular second-order Hamiltonian systems, and our results greatly generalize some well known results such as [1,Theorem 3.6]. Moreover, we exhibit two simple and instructive singular potential examples to make our result more clear, which have not been solved by known results.

Citation: Liang Ding, Rongrong Tian, Jinlong Wei. Nonconstant periodic solutions with any fixed energy for singular Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1617-1625. doi: 10.3934/dcdsb.2018222
##### References:
 [1] A. Ambrosetti and V. C. Zelati, Closed orbits of fixed energy for singular Hamiltonian systems, Arch. Ration. Mech. Anal., 112 (1990), 339-362.  doi: 10.1007/BF02384078.  Google Scholar [2] A. Ambrosetti and V. C. Zelati, Closed orbits of fixed energy for a class of N-body problems, Ann. Institute H. Poincaré-Anal. Non Linéaive, 9 (1992), 187-200.  doi: 10.1016/S0294-1449(16)30244-X.  Google Scholar [3] A. Ambrosetti and V. C. Zelati, Periodic Solutions of Singular Lagrangian Systems, Birkäuser, Boston, 1993. doi: 10.1007/978-1-4612-0319-3.  Google Scholar [4] V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, Springer-Verlag, Berlin, 1997.  Google Scholar [5] V. Benci, Normal modes of a Lagrangian system constrained in a potential well, Ann. Institute H. Poincaré-Anal. Non Linéaive, 1 (1984), 379-400.  doi: 10.1016/S0294-1449(16)30419-X.  Google Scholar [6] V. Benci and P. H. Rabinowitz, Critical point theorems for indefinite functionals, Invent. Math., 52 (1979), 241-273.  doi: 10.1007/BF01389883.  Google Scholar [7] M. Boughariou, Closed orbits of Hamiltonian systems on non-compact prescribed energy surfaces, Discrete Contin. Dynam. Systems, 9 (2003), 603-616.  doi: 10.3934/dcds.2003.9.603.  Google Scholar [8] C. Carminati, E. Sere and K. Tanaka, The fixed energy problem for a class of nonconvex singular Hamiltonian systems, J. Differential Equations, 230 (2006), 362-377.  doi: 10.1016/j.jde.2006.01.021.  Google Scholar [9] R. Castelli, Topologically distinct collision-free periodic solutions for the $N$ -center problem, Arch. Ration. Mech. Anal., 223 (2017), 941-975.  doi: 10.1007/s00205-016-1049-0.  Google Scholar [10] C. F. Che and X. P. Xue, Periodic solutions for second order Hamiltonian systems on an arbitrary energy surface, Ann. Pol. Math., 105 (2012), 1-12.  doi: 10.4064/ap105-1-1.  Google Scholar [11] W. B. Gordon, Conservative dynamical systems involving strong forces, Trans. Am. Math. Soc., 204 (1975), 113-135.  doi: 10.1090/S0002-9947-1975-0377983-1.  Google Scholar [12] Y. M. Long and S. Q. Zhang, Geometric characterization for variational minimization solutions of the 3-body problem with fixed energy, J. Differential Equations, 160 (2000), 422-438.  doi: 10.1006/jdeq.1999.3659.  Google Scholar [13] P. Majer and S. Terracini, Periodic solutions to some N-body type problems: the fixed energy case, Duke Math. J., 69 (1993), 683-697.  doi: 10.1215/S0012-7094-93-06929-3.  Google Scholar [14] P. Majer and S. Terracini, Periodic solutions to some problems of n-body type, Arch. Ration. Mech. Anal., 124 (1993), 381-404.  doi: 10.1007/BF00375608.  Google Scholar [15] J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer-Verlag, New York-Berlin, 1989. doi: 10.1007/978-1-4757-2061-7.  Google Scholar [16] D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems, Springer-Verlag, New York, 2014. doi: 10.1007/978-1-4614-9323-5.  Google Scholar [17] R. S. Palais, The principle of symmetric criticality, Commun. Math. Phys., 69 (1979), 19-30.  doi: 10.1007/BF01941322.  Google Scholar [18] L. Pisani, Periodic solutions with prescribed energy for singular conservative systems involving strong forces, Nonlinear Anal. Theor., 21 (1993), 167-179.  doi: 10.1016/0362-546X(93)90107-4.  Google Scholar [19] P. H. Rabinowitz, Periodic solutions of Hamiltonian systems, Commun. Pur. Appl. Math., 31 (1978), 157-184.  doi: 10.1002/cpa.3160310203.  Google Scholar [20] E. M. Stein and R. Shakarchi, Real Analysis: Measure Theory, Integration, and Hilbert Spaces, Princeton University Press, Princeton, 2005.  Google Scholar [21] S. Terracini, Multiplicity of periodic solution with prescribed energy to singular dynamical systems, Ann. Institute H. Poincaré-Anal. Non Linéaive, 9 (1992), 597-641.  doi: 10.1016/S0294-1449(16)30224-4.  Google Scholar [22] E. Vitillaro, Periodic solutions for singular conservative systems, J. Math. Anal. Appl., 185 (1994), 403-429.  doi: 10.1006/jmaa.1994.1258.  Google Scholar [23] D. L. Wu, C. Li and P. F. Yuan, Periodic solutions for a class of second-order Hamiltonian systems of precribed energy, Electron. J. Qual. Theo., 77 (2015), 1-10.   Google Scholar [24] S. Q. Zhang, Multiple closed orbits of fixed energy for N-body-type problems with gravitational potentials, J. Math. Anal. Appl., 208 (1997), 462-475.  doi: 10.1006/jmaa.1997.5338.  Google Scholar [25] S. Q. Zhang, Periodic solutions for some second order Hamiltonian systems, Nonlinearity, 22 (2009), 2141-2150.  doi: 10.1088/0951-7715/22/9/005.  Google Scholar

show all references

##### References:
 [1] A. Ambrosetti and V. C. Zelati, Closed orbits of fixed energy for singular Hamiltonian systems, Arch. Ration. Mech. Anal., 112 (1990), 339-362.  doi: 10.1007/BF02384078.  Google Scholar [2] A. Ambrosetti and V. C. Zelati, Closed orbits of fixed energy for a class of N-body problems, Ann. Institute H. Poincaré-Anal. Non Linéaive, 9 (1992), 187-200.  doi: 10.1016/S0294-1449(16)30244-X.  Google Scholar [3] A. Ambrosetti and V. C. Zelati, Periodic Solutions of Singular Lagrangian Systems, Birkäuser, Boston, 1993. doi: 10.1007/978-1-4612-0319-3.  Google Scholar [4] V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, Springer-Verlag, Berlin, 1997.  Google Scholar [5] V. Benci, Normal modes of a Lagrangian system constrained in a potential well, Ann. Institute H. Poincaré-Anal. Non Linéaive, 1 (1984), 379-400.  doi: 10.1016/S0294-1449(16)30419-X.  Google Scholar [6] V. Benci and P. H. Rabinowitz, Critical point theorems for indefinite functionals, Invent. Math., 52 (1979), 241-273.  doi: 10.1007/BF01389883.  Google Scholar [7] M. Boughariou, Closed orbits of Hamiltonian systems on non-compact prescribed energy surfaces, Discrete Contin. Dynam. Systems, 9 (2003), 603-616.  doi: 10.3934/dcds.2003.9.603.  Google Scholar [8] C. Carminati, E. Sere and K. Tanaka, The fixed energy problem for a class of nonconvex singular Hamiltonian systems, J. Differential Equations, 230 (2006), 362-377.  doi: 10.1016/j.jde.2006.01.021.  Google Scholar [9] R. Castelli, Topologically distinct collision-free periodic solutions for the $N$ -center problem, Arch. Ration. Mech. Anal., 223 (2017), 941-975.  doi: 10.1007/s00205-016-1049-0.  Google Scholar [10] C. F. Che and X. P. Xue, Periodic solutions for second order Hamiltonian systems on an arbitrary energy surface, Ann. Pol. Math., 105 (2012), 1-12.  doi: 10.4064/ap105-1-1.  Google Scholar [11] W. B. Gordon, Conservative dynamical systems involving strong forces, Trans. Am. Math. Soc., 204 (1975), 113-135.  doi: 10.1090/S0002-9947-1975-0377983-1.  Google Scholar [12] Y. M. Long and S. Q. Zhang, Geometric characterization for variational minimization solutions of the 3-body problem with fixed energy, J. Differential Equations, 160 (2000), 422-438.  doi: 10.1006/jdeq.1999.3659.  Google Scholar [13] P. Majer and S. Terracini, Periodic solutions to some N-body type problems: the fixed energy case, Duke Math. J., 69 (1993), 683-697.  doi: 10.1215/S0012-7094-93-06929-3.  Google Scholar [14] P. Majer and S. Terracini, Periodic solutions to some problems of n-body type, Arch. Ration. Mech. Anal., 124 (1993), 381-404.  doi: 10.1007/BF00375608.  Google Scholar [15] J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer-Verlag, New York-Berlin, 1989. doi: 10.1007/978-1-4757-2061-7.  Google Scholar [16] D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems, Springer-Verlag, New York, 2014. doi: 10.1007/978-1-4614-9323-5.  Google Scholar [17] R. S. Palais, The principle of symmetric criticality, Commun. Math. Phys., 69 (1979), 19-30.  doi: 10.1007/BF01941322.  Google Scholar [18] L. Pisani, Periodic solutions with prescribed energy for singular conservative systems involving strong forces, Nonlinear Anal. Theor., 21 (1993), 167-179.  doi: 10.1016/0362-546X(93)90107-4.  Google Scholar [19] P. H. Rabinowitz, Periodic solutions of Hamiltonian systems, Commun. Pur. Appl. Math., 31 (1978), 157-184.  doi: 10.1002/cpa.3160310203.  Google Scholar [20] E. M. Stein and R. Shakarchi, Real Analysis: Measure Theory, Integration, and Hilbert Spaces, Princeton University Press, Princeton, 2005.  Google Scholar [21] S. Terracini, Multiplicity of periodic solution with prescribed energy to singular dynamical systems, Ann. Institute H. Poincaré-Anal. Non Linéaive, 9 (1992), 597-641.  doi: 10.1016/S0294-1449(16)30224-4.  Google Scholar [22] E. Vitillaro, Periodic solutions for singular conservative systems, J. Math. Anal. Appl., 185 (1994), 403-429.  doi: 10.1006/jmaa.1994.1258.  Google Scholar [23] D. L. Wu, C. Li and P. F. Yuan, Periodic solutions for a class of second-order Hamiltonian systems of precribed energy, Electron. J. Qual. Theo., 77 (2015), 1-10.   Google Scholar [24] S. Q. Zhang, Multiple closed orbits of fixed energy for N-body-type problems with gravitational potentials, J. Math. Anal. Appl., 208 (1997), 462-475.  doi: 10.1006/jmaa.1997.5338.  Google Scholar [25] S. Q. Zhang, Periodic solutions for some second order Hamiltonian systems, Nonlinearity, 22 (2009), 2141-2150.  doi: 10.1088/0951-7715/22/9/005.  Google Scholar
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