American Institute of Mathematical Sciences

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September  2019, 18(5): 2855-2878. doi: 10.3934/cpaa.2019128

Homoclinic orbits for a class of asymptotically quadratic Hamiltonian systems

Ying Lv , Yan-Fang Xue and  Chun-Lei Tang ,

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

* Corresponding author

Received  October 2017 Revised  April 2018 Published  April 2019

Fund Project: Lv is supported by National Natural Science Foundation of China (No.11601438), Xue and Tang are supported by National Natural Science Foundation of China (No.11471267).

In this paper we consider the homoclinic orbits for a class of second order Hamiltonian systems of the form
$ \ddot{q}(t)-\lambda q(t)+\nabla W(t,q(t)) = 0 $
where
$ \lambda>0 $
 is a parameter,
$ \frac{|\nabla W(t,x)|}{|x|} $
 asymptotically tends to a constant as
$ |x|\rightarrow\infty $
 and
$ |t|\rightarrow\infty $
. Via the variational method, two new theorems are proved.
Keywords: Homoclinic orbit, Hamiltonian systems, asymptotically quadratic, variational method, concentration-compactness principle.
Mathematics Subject Classification: Primary: 37J45, 37K05; Secondary: 58E05.
Citation: Ying Lv, Yan-Fang Xue, Chun-Lei Tang. Homoclinic orbits for a class of asymptotically quadratic Hamiltonian systems. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2855-2878. doi: 10.3934/cpaa.2019128
References:
[1]

C. O. Alves, P. C. Carriao and O. H. Miyagaki, Existence of homoclinic orbits for asymptotically periodic systems involving Duffing-like equation, Appl. Math. Lett., 16 (2003), 639-642.  doi: 10.1016/S0893-9659(03)00059-4.  Google Scholar

[2]

G. Arioli and A. Szulkin, Homoclinic solution for a class of systems of second order differential equtions, Tech. Rep. 5, Dept. of Math., Univ. Stockholm, Sweden, 1995. doi: 10.12775/TMNA.1995.040.  Google Scholar

[3]

P. Bartolo, V. Benci and D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with ``strong" resonance at infinity, Nonlinear Anal., 7 (1983), 981-1012.  doi: 10.1016/0362-546X(83)90115-3.  Google Scholar

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[5]

P. C. Carriao and O. H. Miyagaki, Existence of homoclinic solutions for a class of time-dependent Hamiltonian systems, J. Math. Anal. Appl., 230 (1999), 157-172.  doi: 10.1006/jmaa.1998.6184.  Google Scholar

[6]

G. W. Chen, Superquadratic or asymptotically quadratic Hamiltonian systems: ground state homoclinic orbits, Ann. Mat. Pura Appl., 194 (2015), 903-918.  doi: 10.1007/s10231-014-0403-9.  Google Scholar

[7]

D. G. Coata and C. A. Magalhães, Variational elliptic problems which are nonquadratic at infinity, Nonlinear Anal., 23 (1994), 1401-1412.  doi: 10.1016/0362-546X(94)90135-X.  Google Scholar

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D. G. Costa and H. Tehrani, On a class of asymptotically linear elliptic problems in $R^{N}$, J. Differential Equations, 173 (2001), 470-494.  doi: 10.1006/jdeq.2000.3944.  Google Scholar

[9]

V. Coti-Zelati, I. Ekeland and E. Sere, A variational approach to homoclinic orbits in Hamiltonian systems, Math. Ann., 288 (1990), 133-160.  doi: 10.1007/BF01444526.  Google Scholar

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Y. Ding, Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems, Nonlinear Anal., 25 (1995), 1095-1113.  doi: 10.1016/0362-546X(94)00229-B.  Google Scholar

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Y. Ding and C. Lee, Homoclinics for asymptotically quadratic and superquadratic Hamiltonian systems, Nonlinear Anal., 71 (2009), 1395-1413.  doi: 10.1016/j.na.2008.10.116.  Google Scholar

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G. H. Fei, The existence of homoclinic orbits for Hamiltonian systems with the potentials changing sign, Chinese Ann. Math. Ser. A, 17 (1996), 403-410.   Google Scholar

[13]

P. L. Felmer and E. A. De B. e. Silva, Homoclinic and periodic orbits for Hamiltonian systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 285-301.   Google Scholar

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P. Korman and A. C. Lazer, Homoclinic orbits for a class of symmetric Hamiltonian systems, Electronic J. Differential Equations, 1994 (1994), 1-10.   Google Scholar

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P. L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case. Ⅰ., Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145.   Google Scholar

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Z. Liu, S. Guo and Z. Zhang, Homoclinic orbits for the second-order Hamiltonian systems, Nonlinear Anal. Real World Appl., 36 (2017), 116-138.  doi: 10.1016/j.nonrwa.2016.12.006.  Google Scholar

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X. Lv, S. Lu and P. Yan, Existence of homoclinic solutions for a class of second-order Hamiltonian systems, Nonlinear Anal., 72 (2010), 390-398.  doi: 10.1016/j.na.2009.06.073.  Google Scholar

[19]

Y. Lv and C. L. Tang, Existence of even homoclinic orbits for second-order Hamiltonian systems, Nonlinear Anal., 67 (2007), 2189-2198.  doi: 10.1016/j.na.2006.08.043.  Google Scholar

[20]

Y. Lv and C. L. Tang, Homoclinic orbits for second-order Hamiltonian systems with subquadratic potentials, Chaos Solitons Fractals, 57 (2013), 137-145.  doi: 10.1016/j.chaos.2013.09.007.  Google Scholar

[21]

I. Marek and J. Joanna, Homoclinic solutions for a class of second order Hamiltonian systems, J. Differential. Equations, 219 (2005), 375-389.  doi: 10.1016/j.jde.2005.06.029.  Google Scholar

[22]

W. Omana and M. Willem, Homoclinic orbits for a class of Hamiltonian systems, Differential Integral Equations, 5 (1992), 1115-1120.   Google Scholar

[23]

Z. Q. Ou and C. L. Tang, Existence of homoclinic solution for the second order Hamiltonian systems, J. Math. Anal. Appl., 291 (2004), 203-213.  doi: 10.1016/j.jmaa.2003.10.026.  Google Scholar

[24]

P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, in CBMS Regional Conference Series in Mathematics, 65. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986. doi: 10.1090/cbms/065.  Google Scholar

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P. H. Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems, Proc. Roy. Soc. Edinburgh Sect. A, 114 (1990), 33-38.  doi: 10.1017/S0308210500024240.  Google Scholar

[26]

P. H. Rabinowitz and K. Tanaka, Some results on connecting orbits for a class of Hamiltonian systems, Math. Z., 206 (1991), 473-499.  doi: 10.1007/BF02571356.  Google Scholar

[27]

A. Salvatore, Homoclinic orbits for a special class of nonautonomous Hamiltonian systems, Nonlinear Anal., 30 (1997), 4849-4857.  doi: 10.1016/S0362-546X(97)00142-9.  Google Scholar

[28]

E. Serra, M. Tarallo and S. Terracini, Subharmonic solutions to second-order differential equations with periodic nonlinearities, Nonlinear Anal., 41 (2000), 649-667.  doi: 10.1016/S0362-546X(98)00302-2.  Google Scholar

[29]

J. Sun, H. Chen and Juan J. Nieto, Homoclinic solutions for a class of subquadratic second-order Hamiltonian systems, J. Math. Anal. Appl., 373 (2011), 20-29.  doi: 10.1016/j.jmaa.2010.06.038.  Google Scholar

[30]

M. Yang and Z. Han, Infinitly homoclinic solutions for second-order Hamiltonian systems with odd nonlinearities, Nonlinear Anal., 74 (2011), 2635-2646.  doi: 10.1016/j.na.2010.12.019.  Google Scholar

[31]

Y. Ye and C. L. Tang, Multiple homoclinic solutions for second-order perturbed Hamiltonian systems, Stud. Appl. Math., 132 (2014), 112-137.  doi: 10.1111/sapm.12023.  Google Scholar

[32]

Z. Zhang and R. Yuan, Homoclinic solutions for a class of non-autonomous subquadratic second-order Hamiltonian systems, Nonlinear Anal., 71 (2009), 4125-4130.  doi: 10.1016/j.na.2009.02.071.  Google Scholar

[33]

Q. Zhang and X. H. Tang, Existence of homoclinic solutions for a class of asymptotically quadratic non-autonomous Hamiltonian systems, Math. Nachr., 285 (2012), 778-789.  doi: 10.1002/mana.201000096.  Google Scholar

[34]

Q. Zheng, Homoclinic solutions for a second-order nonperiodic asymptotically linear Hamiltonian systems, Abstr. Appl. Anal., 7 (2013), 34-37.  doi: 10.1155/2013/417020.  Google Scholar

[35]

W. Zou and S. Li, Infinitely many homoclinic orbits for the second-order Hamiltonian systems, Appl. Math. Lett., 16 (2003), 1283-1287.  doi: 10.1016/S0893-9659(03)90130-3.  Google Scholar

show all references

References:
[1]

C. O. Alves, P. C. Carriao and O. H. Miyagaki, Existence of homoclinic orbits for asymptotically periodic systems involving Duffing-like equation, Appl. Math. Lett., 16 (2003), 639-642.  doi: 10.1016/S0893-9659(03)00059-4.  Google Scholar

[2]

G. Arioli and A. Szulkin, Homoclinic solution for a class of systems of second order differential equtions, Tech. Rep. 5, Dept. of Math., Univ. Stockholm, Sweden, 1995. doi: 10.12775/TMNA.1995.040.  Google Scholar

[3]

P. Bartolo, V. Benci and D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with ``strong" resonance at infinity, Nonlinear Anal., 7 (1983), 981-1012.  doi: 10.1016/0362-546X(83)90115-3.  Google Scholar

[4]

F. A. Berezin and M. A. Shubin, The Schrodinger Equation, Kluwer, Dordrecht, 1991. doi: 10.1007/978-94-011-3154-4.  Google Scholar

[5]

P. C. Carriao and O. H. Miyagaki, Existence of homoclinic solutions for a class of time-dependent Hamiltonian systems, J. Math. Anal. Appl., 230 (1999), 157-172.  doi: 10.1006/jmaa.1998.6184.  Google Scholar

[6]

G. W. Chen, Superquadratic or asymptotically quadratic Hamiltonian systems: ground state homoclinic orbits, Ann. Mat. Pura Appl., 194 (2015), 903-918.  doi: 10.1007/s10231-014-0403-9.  Google Scholar

[7]

D. G. Coata and C. A. Magalhães, Variational elliptic problems which are nonquadratic at infinity, Nonlinear Anal., 23 (1994), 1401-1412.  doi: 10.1016/0362-546X(94)90135-X.  Google Scholar

[8]

D. G. Costa and H. Tehrani, On a class of asymptotically linear elliptic problems in $R^{N}$, J. Differential Equations, 173 (2001), 470-494.  doi: 10.1006/jdeq.2000.3944.  Google Scholar

[9]

V. Coti-Zelati, I. Ekeland and E. Sere, A variational approach to homoclinic orbits in Hamiltonian systems, Math. Ann., 288 (1990), 133-160.  doi: 10.1007/BF01444526.  Google Scholar

[10]

Y. Ding, Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems, Nonlinear Anal., 25 (1995), 1095-1113.  doi: 10.1016/0362-546X(94)00229-B.  Google Scholar

[11]

Y. Ding and C. Lee, Homoclinics for asymptotically quadratic and superquadratic Hamiltonian systems, Nonlinear Anal., 71 (2009), 1395-1413.  doi: 10.1016/j.na.2008.10.116.  Google Scholar

[12]

G. H. Fei, The existence of homoclinic orbits for Hamiltonian systems with the potentials changing sign, Chinese Ann. Math. Ser. A, 17 (1996), 403-410.   Google Scholar

[13]

P. L. Felmer and E. A. De B. e. Silva, Homoclinic and periodic orbits for Hamiltonian systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 285-301.   Google Scholar

[14]

P. Korman and A. C. Lazer, Homoclinic orbits for a class of symmetric Hamiltonian systems, Electronic J. Differential Equations, 1994 (1994), 1-10.   Google Scholar

[15]

P. L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case. Ⅰ., Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145.   Google Scholar

[16]

P. L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case. Ⅱ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283.   Google Scholar

[17]

Z. Liu, S. Guo and Z. Zhang, Homoclinic orbits for the second-order Hamiltonian systems, Nonlinear Anal. Real World Appl., 36 (2017), 116-138.  doi: 10.1016/j.nonrwa.2016.12.006.  Google Scholar

[18]

X. Lv, S. Lu and P. Yan, Existence of homoclinic solutions for a class of second-order Hamiltonian systems, Nonlinear Anal., 72 (2010), 390-398.  doi: 10.1016/j.na.2009.06.073.  Google Scholar

[19]

Y. Lv and C. L. Tang, Existence of even homoclinic orbits for second-order Hamiltonian systems, Nonlinear Anal., 67 (2007), 2189-2198.  doi: 10.1016/j.na.2006.08.043.  Google Scholar

[20]

Y. Lv and C. L. Tang, Homoclinic orbits for second-order Hamiltonian systems with subquadratic potentials, Chaos Solitons Fractals, 57 (2013), 137-145.  doi: 10.1016/j.chaos.2013.09.007.  Google Scholar

[21]

I. Marek and J. Joanna, Homoclinic solutions for a class of second order Hamiltonian systems, J. Differential. Equations, 219 (2005), 375-389.  doi: 10.1016/j.jde.2005.06.029.  Google Scholar

[22]

W. Omana and M. Willem, Homoclinic orbits for a class of Hamiltonian systems, Differential Integral Equations, 5 (1992), 1115-1120.   Google Scholar

[23]

Z. Q. Ou and C. L. Tang, Existence of homoclinic solution for the second order Hamiltonian systems, J. Math. Anal. Appl., 291 (2004), 203-213.  doi: 10.1016/j.jmaa.2003.10.026.  Google Scholar

[24]

P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, in CBMS Regional Conference Series in Mathematics, 65. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986. doi: 10.1090/cbms/065.  Google Scholar

[25]

P. H. Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems, Proc. Roy. Soc. Edinburgh Sect. A, 114 (1990), 33-38.  doi: 10.1017/S0308210500024240.  Google Scholar

[26]

P. H. Rabinowitz and K. Tanaka, Some results on connecting orbits for a class of Hamiltonian systems, Math. Z., 206 (1991), 473-499.  doi: 10.1007/BF02571356.  Google Scholar

[27]

A. Salvatore, Homoclinic orbits for a special class of nonautonomous Hamiltonian systems, Nonlinear Anal., 30 (1997), 4849-4857.  doi: 10.1016/S0362-546X(97)00142-9.  Google Scholar

[28]

E. Serra, M. Tarallo and S. Terracini, Subharmonic solutions to second-order differential equations with periodic nonlinearities, Nonlinear Anal., 41 (2000), 649-667.  doi: 10.1016/S0362-546X(98)00302-2.  Google Scholar

[29]

J. Sun, H. Chen and Juan J. Nieto, Homoclinic solutions for a class of subquadratic second-order Hamiltonian systems, J. Math. Anal. Appl., 373 (2011), 20-29.  doi: 10.1016/j.jmaa.2010.06.038.  Google Scholar

[30]

M. Yang and Z. Han, Infinitly homoclinic solutions for second-order Hamiltonian systems with odd nonlinearities, Nonlinear Anal., 74 (2011), 2635-2646.  doi: 10.1016/j.na.2010.12.019.  Google Scholar

[31]

Y. Ye and C. L. Tang, Multiple homoclinic solutions for second-order perturbed Hamiltonian systems, Stud. Appl. Math., 132 (2014), 112-137.  doi: 10.1111/sapm.12023.  Google Scholar

[32]

Z. Zhang and R. Yuan, Homoclinic solutions for a class of non-autonomous subquadratic second-order Hamiltonian systems, Nonlinear Anal., 71 (2009), 4125-4130.  doi: 10.1016/j.na.2009.02.071.  Google Scholar

[33]

Q. Zhang and X. H. Tang, Existence of homoclinic solutions for a class of asymptotically quadratic non-autonomous Hamiltonian systems, Math. Nachr., 285 (2012), 778-789.  doi: 10.1002/mana.201000096.  Google Scholar

[34]

Q. Zheng, Homoclinic solutions for a second-order nonperiodic asymptotically linear Hamiltonian systems, Abstr. Appl. Anal., 7 (2013), 34-37.  doi: 10.1155/2013/417020.  Google Scholar

[35]

W. Zou and S. Li, Infinitely many homoclinic orbits for the second-order Hamiltonian systems, Appl. Math. Lett., 16 (2003), 1283-1287.  doi: 10.1016/S0893-9659(03)90130-3.  Google Scholar

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