-
Previous Article
A SIS reaction-diffusion model with a free boundary condition and nonhomogeneous coefficients
- DCDS-B Home
- This Issue
-
Next Article
Novel spectral methods for Schrödinger equations with an inverse square potential on the whole space
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 |
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 [
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. |
[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. |
[3] |
A. Ambrosetti and V. C. Zelati,
Periodic Solutions of Singular Lagrangian Systems, Birkäuser, Boston, 1993.
doi: 10.1007/978-1-4612-0319-3. |
[4] |
V. I. Arnold, V. V. Kozlov and A. I. Neishtadt,
Mathematical Aspects of Classical and Celestial Mechanics, Springer-Verlag, Berlin, 1997. |
[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. |
[6] |
V. Benci and P. H. Rabinowitz,
Critical point theorems for indefinite functionals, Invent. Math., 52 (1979), 241-273.
doi: 10.1007/BF01389883. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[17] |
R. S. Palais,
The principle of symmetric criticality, Commun. Math. Phys., 69 (1979), 19-30.
doi: 10.1007/BF01941322. |
[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. |
[19] |
P. H. Rabinowitz,
Periodic solutions of Hamiltonian systems, Commun. Pur. Appl. Math., 31 (1978), 157-184.
doi: 10.1002/cpa.3160310203. |
[20] |
E. M. Stein and R. Shakarchi,
Real Analysis: Measure Theory, Integration, and Hilbert Spaces, Princeton University Press, Princeton, 2005. |
[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. |
[22] |
E. Vitillaro,
Periodic solutions for singular conservative systems, J. Math. Anal. Appl., 185 (1994), 403-429.
doi: 10.1006/jmaa.1994.1258. |
[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.
|
[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. |
[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. |
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. |
[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. |
[3] |
A. Ambrosetti and V. C. Zelati,
Periodic Solutions of Singular Lagrangian Systems, Birkäuser, Boston, 1993.
doi: 10.1007/978-1-4612-0319-3. |
[4] |
V. I. Arnold, V. V. Kozlov and A. I. Neishtadt,
Mathematical Aspects of Classical and Celestial Mechanics, Springer-Verlag, Berlin, 1997. |
[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. |
[6] |
V. Benci and P. H. Rabinowitz,
Critical point theorems for indefinite functionals, Invent. Math., 52 (1979), 241-273.
doi: 10.1007/BF01389883. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[17] |
R. S. Palais,
The principle of symmetric criticality, Commun. Math. Phys., 69 (1979), 19-30.
doi: 10.1007/BF01941322. |
[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. |
[19] |
P. H. Rabinowitz,
Periodic solutions of Hamiltonian systems, Commun. Pur. Appl. Math., 31 (1978), 157-184.
doi: 10.1002/cpa.3160310203. |
[20] |
E. M. Stein and R. Shakarchi,
Real Analysis: Measure Theory, Integration, and Hilbert Spaces, Princeton University Press, Princeton, 2005. |
[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. |
[22] |
E. Vitillaro,
Periodic solutions for singular conservative systems, J. Math. Anal. Appl., 185 (1994), 403-429.
doi: 10.1006/jmaa.1994.1258. |
[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.
|
[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. |
[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. |
[1] |
Mitsuru Shibayama. Periodic solutions for a prescribed-energy problem of singular Hamiltonian systems. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2705-2715. doi: 10.3934/dcds.2017116 |
[2] |
Tianqing An, Zhi-Qiang Wang. Periodic solutions of Hamiltonian systems with anisotropic growth. Communications on Pure and Applied Analysis, 2010, 9 (4) : 1069-1082. doi: 10.3934/cpaa.2010.9.1069 |
[3] |
Alessandro Fonda, Andrea Sfecci. Multiple periodic solutions of Hamiltonian systems confined in a box. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1425-1436. doi: 10.3934/dcds.2017059 |
[4] |
Anna Capietto, Walter Dambrosio, Tiantian Ma, Zaihong Wang. Unbounded solutions and periodic solutions of perturbed isochronous Hamiltonian systems at resonance. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 1835-1856. doi: 10.3934/dcds.2013.33.1835 |
[5] |
Josep M. Olm, Xavier Ros-Oton. Existence of periodic solutions with nonconstant sign in a class of generalized Abel equations. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1603-1614. doi: 10.3934/dcds.2013.33.1603 |
[6] |
Jianshe Yu, Honghua Bin, Zhiming Guo. Periodic solutions for discrete convex Hamiltonian systems via Clarke duality. Discrete and Continuous Dynamical Systems, 2006, 15 (3) : 939-950. doi: 10.3934/dcds.2006.15.939 |
[7] |
Pietro-Luciano Buono, Daniel C. Offin. Instability criterion for periodic solutions with spatio-temporal symmetries in Hamiltonian systems. Journal of Geometric Mechanics, 2017, 9 (4) : 439-457. doi: 10.3934/jgm.2017017 |
[8] |
Shiwang Ma. Nontrivial periodic solutions for asymptotically linear hamiltonian systems at resonance. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2361-2380. doi: 10.3934/cpaa.2013.12.2361 |
[9] |
Laura Olian Fannio. Multiple periodic solutions of Hamiltonian systems with strong resonance at infinity. Discrete and Continuous Dynamical Systems, 1997, 3 (2) : 251-264. doi: 10.3934/dcds.1997.3.251 |
[10] |
Paolo Gidoni, Alessandro Margheri. Lower bound on the number of periodic solutions for asymptotically linear planar Hamiltonian systems. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 585-606. doi: 10.3934/dcds.2019024 |
[11] |
Giuseppe Cordaro. Existence and location of periodic solutions to convex and non coercive Hamiltonian systems. Discrete and Continuous Dynamical Systems, 2005, 12 (5) : 983-996. doi: 10.3934/dcds.2005.12.983 |
[12] |
Juhong Kuang, Weiyi Chen, Zhiming Guo. Periodic solutions with prescribed minimal period for second order even Hamiltonian systems. Communications on Pure and Applied Analysis, 2022, 21 (1) : 47-59. doi: 10.3934/cpaa.2021166 |
[13] |
Ernest Fontich, Rafael de la Llave, Yannick Sire. A method for the study of whiskered quasi-periodic and almost-periodic solutions in finite and infinite dimensional Hamiltonian systems. Electronic Research Announcements, 2009, 16: 9-22. doi: 10.3934/era.2009.16.9 |
[14] |
V. Barbu. Periodic solutions to unbounded Hamiltonian system. Discrete and Continuous Dynamical Systems, 1995, 1 (2) : 277-283. doi: 10.3934/dcds.1995.1.277 |
[15] |
Juan Campos, Rafael Ortega. Location of fixed points and periodic solutions in the plane. Discrete and Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 517-523. doi: 10.3934/dcdsb.2008.9.517 |
[16] |
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 |
[17] |
Chungen Liu, Xiaofei Zhang. Subharmonic solutions and minimal periodic solutions of first-order Hamiltonian systems with anisotropic growth. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1559-1574. doi: 10.3934/dcds.2017064 |
[18] |
Xingyong Zhang, Xianhua Tang. Some united existence results of periodic solutions for non-quadratic second order Hamiltonian systems. Communications on Pure and Applied Analysis, 2014, 13 (1) : 75-95. doi: 10.3934/cpaa.2014.13.75 |
[19] |
Jean Mawhin. Periodic solutions of second order Lagrangian difference systems with bounded or singular $\phi$-Laplacian and periodic potential. Discrete and Continuous Dynamical Systems - S, 2013, 6 (4) : 1065-1076. doi: 10.3934/dcdss.2013.6.1065 |
[20] |
Jia Li, Junxiang Xu. On the reducibility of a class of almost periodic Hamiltonian systems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3905-3919. doi: 10.3934/dcdsb.2020268 |
2020 Impact Factor: 1.327
Tools
Metrics
Other articles
by authors
[Back to Top]