March  2017, 37(3): 1559-1574. doi: 10.3934/dcds.2017064

Subharmonic solutions and minimal periodic solutions of first-order Hamiltonian systems with anisotropic growth

School of Mathematics and LPMC, Nankai University, Tianjin 300071, China

* Corresponding author: Chungen Liu

Received  June 2016 Revised  October 2016 Published  December 2016

Fund Project: The first author is supported partially by the NSF of China (11071127,10621101), 973 Program of MOST (2011CB808002) and SRFDP.

Using a homologically link theorem in variational theory and iteration inequalities of Maslov-type index, we show the existence of a sequence of subharmonic solutions of non-autonomous Hamiltonian systems with the Hamiltonian functions satisfying some anisotropic growth conditions, i.e., the Hamiltonian functions may have simultaneously, in different components, superquadratic, subquadratic and quadratic behaviors. Moreover, we also consider the minimal period problem of some autonomous Hamiltonian systems with anisotropic growth.

Citation: Chungen Liu, Xiaofei Zhang. Subharmonic solutions and minimal periodic solutions of first-order Hamiltonian systems with anisotropic growth. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1559-1574. doi: 10.3934/dcds.2017064
References:
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A. Abbondandolo, Morse Theory for Hamiltonian Systems Chapman, Hall, London, 2001.  Google Scholar

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T. An and Z. Wang, Periodic solutions of Hamiltonian systems with anisotropic growth, Commun. Pure Appl. Anal., 9 (2010), 1069-1082.  doi: 10.3934/cpaa.2010.9.1069.  Google Scholar

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S. Chen and C. 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.  Google Scholar

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D. Dong and Y. Long, The iteration formula of the Maslov-type index theory with applications to nonlinear Hamiltonian systems, Trans. Amer. Math. Soc., 349 (1997), 2619-2661.  doi: 10.1090/S0002-9947-97-01718-2.  Google Scholar

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I. Ekeland and H. Hofer, Subharmonics of convex nonautonomous Hamiltonian systems, Comm. Pure Appl. Math., 40 (1987), 1-36.  doi: 10.1002/cpa.3160400102.  Google Scholar

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G. Fei, On periodic solutions of superquadratic Hamiltonian systems, Electron. J. Differential Equations, 8 (2002), 1-12.   Google Scholar

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G. FeiS. Kim and T. Wang, Minimal period estimates of periodic solutions for superquadratic Hamiltonian systmes, J. Math. Anal. Appl., 238 (1999), 216-233.  doi: 10.1006/jmaa.1999.6527.  Google Scholar

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P. L. Felmer, Periodic solutions of ''superquadratic'' Hamiltonian systems, J. Differential Equations, 102 (1993), 188-207.  doi: 10.1006/jdeq.1993.1027.  Google Scholar

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C. Li, Brake subharmonic solutions of subquadratic Hamiltonian systems, Chin. Ann. Math. Ser. B, 37 (2016), 405-418.  doi: 10.1007/s11401-016-0970-8.  Google Scholar

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C. Li, The study of minimal period estimates for brake orbits of autonomous subquadratic Hamiltonian systems, Acta Math. Sin. (Engl. Ser.), 31 (2015), 1645-1658.  doi: 10.1007/s10114-015-4421-3.  Google Scholar

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C. Li and C. Liu, Brake subharmonic solutions of first order Hamiltonian systems, Sci. China Math., 53 (2010), 2719-2732.  doi: 10.1007/s11425-010-4105-5.  Google Scholar

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C. LiZ. Ou and C. Tang, Periodic and subharmonic solutions for a class of non-autonomous Hamiltonian systems, Nonlinear Anal., 75 (2012), 2262-2272.  doi: 10.1016/j.na.2011.10.026.  Google Scholar

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C. Liu, Subharmonic solutions of Hamiltonian systems, Nonlinear Anal., 42 (2000), 185-198.  doi: 10.1016/S0362-546X(98)00339-3.  Google Scholar

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C. Liu, Minimal period estimates for brake orbits of nonlinear symmetric Hamiltonian systems, Discrete Contin. Dyn. Syst., 27 (2010), 337-355.  doi: 10.3934/dcds.2010.27.337.  Google Scholar

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C. Liu, {Relative index theories and applications}, preprint. Google Scholar

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C. Liu and Y. Long, Iteration inequalities of the Maslov-type index theory with applications, J. Differential Equations, 165 (2000), 355-376.  doi: 10.1006/jdeq.2000.3775.  Google Scholar

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C. Liu and S. Tang, Subharmonic P-solutions of first order Hamiltonian systems, preprint. Google Scholar

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C. Liu and S. Tang, Iteration inequalities of the Maslov P-index theory with applications, Nonlinear Anal., 127 (2015), 215-234.  doi: 10.1016/j.na.2015.06.029.  Google Scholar

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Y. Long, Index Theory for Symplectic Paths with Applications Birkhauser Verlag Basel · Boston · Berlin, 2002. doi: 10.1007/978-3-0348-8175-3.  Google Scholar

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

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R. Michalek and G. Tarantello, Subharmonic solutions with prescribed minimal period for nonautonomous Hamiltonian systems, J. Differential Equations, 72 (1988), 28-55.  doi: 10.1016/0022-0396(88)90148-9.  Google Scholar

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K. Perera and M. Schechter, Topics in Critical Point Theory Cambridge University Press, 2013.  Google Scholar

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

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P. H. Rabinowitz, On subhamonic solutions of Hamiltonian systems, Comm. Pure Appl. Math., 33 (1980), 609-633.  doi: 10.1002/cpa.3160330504.  Google Scholar

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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.  Google Scholar

[31]

E. Silva, Subharmonic solutions for subquadratic Hamiltonian systems, J. Differential Equations, 115 (1995), 120-145.  doi: 10.1006/jdeq.1995.1007.  Google Scholar

[32]

Q. XingF. Guo and X. Zhang, One generalized critical point theorem and its applications on super-quadratic Hamiltonian systems, Taiwanese J. Math., 20 (2016), 1093-1116.  doi: 10.11650/tjm.20.2016.7128.  Google Scholar

[33]

D. Zhang, Symmetric period solutions with prescribed minimal period for even autonomous semipositive Hamiltonian systems, Sci. China Math., 57 (2014), 81-96.   Google Scholar

[34]

X. 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.  Google Scholar

[35]

X. Zhang and F. Guo, Multiplicity of Subharmonic Solutions and Periodic Solutions of a Particular Type of Super-quadratic Hamiltonian Systems, Commun. Pure Appl. Anal., 15 (2016), 1625-1642.  doi: 10.3934/cpaa.2016005.  Google Scholar

show all references

References:
[1]

A. Abbondandolo, Morse Theory for Hamiltonian Systems Chapman, Hall, London, 2001.  Google Scholar

[2]

T. An and Z. Wang, Periodic solutions of Hamiltonian systems with anisotropic growth, Commun. Pure Appl. Anal., 9 (2010), 1069-1082.  doi: 10.3934/cpaa.2010.9.1069.  Google Scholar

[3]

K. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems Birkh-äuser, Boston, 1993. doi: 10.1007/978-1-4612-0385-8.  Google Scholar

[4]

S. Chen and C. 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.  Google Scholar

[5]

D. Dong and Y. Long, The iteration formula of the Maslov-type index theory with applications to nonlinear Hamiltonian systems, Trans. Amer. Math. Soc., 349 (1997), 2619-2661.  doi: 10.1090/S0002-9947-97-01718-2.  Google Scholar

[6]

I. Ekeland, Convexity Methods in Hamiltonian Mechanics Springer, Berlin, 1990. doi: 10.1007/978-3-642-74331-3.  Google Scholar

[7]

I. Ekeland and H. Hofer, Subharmonics of convex nonautonomous Hamiltonian systems, Comm. Pure Appl. Math., 40 (1987), 1-36.  doi: 10.1002/cpa.3160400102.  Google Scholar

[8]

G. Fei, Relative morse index and its application to Hamiltonian systems in the presence of symmetries, J. Differential Equations, 122 (1995), 302-315.  doi: 10.1006/jdeq.1995.1150.  Google Scholar

[9]

G. Fei, On periodic solutions of superquadratic Hamiltonian systems, Electron. J. Differential Equations, 8 (2002), 1-12.   Google Scholar

[10]

G. Fei and Q. Qiu, Periodic solutions of asymptotically linear Hamiltonian systems, Chinese Ann. Math. Ser. B, 18 (1997), 359-372.   Google Scholar

[11]

G. Fei and Q. Qiu, Minimal period solutions of nonlinear Hamiltonian systems, Nonlinear Anal., 27 (1996), 821-839.  doi: 10.1016/0362-546X(95)00077-9.  Google Scholar

[12]

G. FeiS. Kim and T. Wang, Minimal period estimates of periodic solutions for superquadratic Hamiltonian systmes, J. Math. Anal. Appl., 238 (1999), 216-233.  doi: 10.1006/jmaa.1999.6527.  Google Scholar

[13]

P. L. Felmer, Periodic solutions of ''superquadratic'' Hamiltonian systems, J. Differential Equations, 102 (1993), 188-207.  doi: 10.1006/jdeq.1993.1027.  Google Scholar

[14]

C. Li, Brake subharmonic solutions of subquadratic Hamiltonian systems, Chin. Ann. Math. Ser. B, 37 (2016), 405-418.  doi: 10.1007/s11401-016-0970-8.  Google Scholar

[15]

C. Li, The study of minimal period estimates for brake orbits of autonomous subquadratic Hamiltonian systems, Acta Math. Sin. (Engl. Ser.), 31 (2015), 1645-1658.  doi: 10.1007/s10114-015-4421-3.  Google Scholar

[16]

C. Li and C. Liu, Brake subharmonic solutions of first order Hamiltonian systems, Sci. China Math., 53 (2010), 2719-2732.  doi: 10.1007/s11425-010-4105-5.  Google Scholar

[17]

C. LiZ. Ou and C. Tang, Periodic and subharmonic solutions for a class of non-autonomous Hamiltonian systems, Nonlinear Anal., 75 (2012), 2262-2272.  doi: 10.1016/j.na.2011.10.026.  Google Scholar

[18]

C. Liu, Subharmonic solutions of Hamiltonian systems, Nonlinear Anal., 42 (2000), 185-198.  doi: 10.1016/S0362-546X(98)00339-3.  Google Scholar

[19]

C. Liu, Minimal period estimates for brake orbits of nonlinear symmetric Hamiltonian systems, Discrete Contin. Dyn. Syst., 27 (2010), 337-355.  doi: 10.3934/dcds.2010.27.337.  Google Scholar

[20]

C. Liu, {Relative index theories and applications}, preprint. Google Scholar

[21]

C. Liu and Y. Long, Iteration inequalities of the Maslov-type index theory with applications, J. Differential Equations, 165 (2000), 355-376.  doi: 10.1006/jdeq.2000.3775.  Google Scholar

[22]

C. Liu and S. Tang, Subharmonic P-solutions of first order Hamiltonian systems, preprint. Google Scholar

[23]

C. Liu and S. Tang, Iteration inequalities of the Maslov P-index theory with applications, Nonlinear Anal., 127 (2015), 215-234.  doi: 10.1016/j.na.2015.06.029.  Google Scholar

[24]

Y. Long, Index Theory for Symplectic Paths with Applications Birkhauser Verlag Basel · Boston · Berlin, 2002. doi: 10.1007/978-3-0348-8175-3.  Google Scholar

[25]

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

[26]

R. Michalek and G. Tarantello, Subharmonic solutions with prescribed minimal period for nonautonomous Hamiltonian systems, J. Differential Equations, 72 (1988), 28-55.  doi: 10.1016/0022-0396(88)90148-9.  Google Scholar

[27]

K. Perera and M. Schechter, Topics in Critical Point Theory Cambridge University Press, 2013.  Google Scholar

[28]

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

[29]

P. H. Rabinowitz, On subhamonic solutions of Hamiltonian systems, Comm. Pure Appl. Math., 33 (1980), 609-633.  doi: 10.1002/cpa.3160330504.  Google Scholar

[30]

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.  Google Scholar

[31]

E. Silva, Subharmonic solutions for subquadratic Hamiltonian systems, J. Differential Equations, 115 (1995), 120-145.  doi: 10.1006/jdeq.1995.1007.  Google Scholar

[32]

Q. XingF. Guo and X. Zhang, One generalized critical point theorem and its applications on super-quadratic Hamiltonian systems, Taiwanese J. Math., 20 (2016), 1093-1116.  doi: 10.11650/tjm.20.2016.7128.  Google Scholar

[33]

D. Zhang, Symmetric period solutions with prescribed minimal period for even autonomous semipositive Hamiltonian systems, Sci. China Math., 57 (2014), 81-96.   Google Scholar

[34]

X. 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.  Google Scholar

[35]

X. Zhang and F. Guo, Multiplicity of Subharmonic Solutions and Periodic Solutions of a Particular Type of Super-quadratic Hamiltonian Systems, Commun. Pure Appl. Anal., 15 (2016), 1625-1642.  doi: 10.3934/cpaa.2016005.  Google Scholar

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