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:
[1]

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

[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.

[3]

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

[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.

[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.

[6]

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

[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.

[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.

[9]

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

[10]

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

[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.

[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.

[13]

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

[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.

[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.

[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.

[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.

[18]

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

[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.

[20]

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

[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.

[22]

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

[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.

[24]

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

[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.

[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.

[27]

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

[28]

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

[29]

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

[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.

[31]

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

[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.

[33]

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

[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.

[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.

show all references

References:
[1]

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

[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.

[3]

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

[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.

[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.

[6]

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

[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.

[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.

[9]

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

[10]

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

[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.

[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.

[13]

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

[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.

[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.

[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.

[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.

[18]

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

[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.

[20]

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

[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.

[22]

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

[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.

[24]

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

[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.

[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.

[27]

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

[28]

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

[29]

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

[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.

[31]

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

[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.

[33]

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

[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.

[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.

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