Homoclinic orbits for a class of asymptotically quadratic Hamiltonian systems

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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.
[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.
[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.
[4]	F. A. Berezin and M. A. Shubin, The Schrodinger Equation, Kluwer, Dordrecht, 1991. doi: 10.1007/978-94-011-3154-4.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[14]	P. Korman and A. C. Lazer, Homoclinic orbits for a class of symmetric Hamiltonian systems, Electronic J. Differential Equations, 1994 (1994), 1-10.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[22]	W. Omana and M. Willem, Homoclinic orbits for a class of Hamiltonian systems, Differential Integral Equations, 5 (1992), 1115-1120.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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.
[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.
[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.
[4]	F. A. Berezin and M. A. Shubin, The Schrodinger Equation, Kluwer, Dordrecht, 1991. doi: 10.1007/978-94-011-3154-4.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[14]	P. Korman and A. C. Lazer, Homoclinic orbits for a class of symmetric Hamiltonian systems, Electronic J. Differential Equations, 1994 (1994), 1-10.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[22]	W. Omana and M. Willem, Homoclinic orbits for a class of Hamiltonian systems, Differential Integral Equations, 5 (1992), 1115-1120.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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