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

Homoclinic orbits for a class of asymptotically quadratic Hamiltonian systems

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.
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. AlvesP. 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. BartoloV. 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-ZelatiI. 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. LiuS. 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. LvS. 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. SerraM. 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. SunH. 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. AlvesP. 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. BartoloV. 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-ZelatiI. 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. LiuS. 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. LvS. 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. SerraM. 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. SunH. 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

[1]

Jun Wang, Junxiang Xu, Fubao Zhang. Homoclinic orbits for a class of Hamiltonian systems with superquadratic or asymptotically quadratic potentials. Communications on Pure & Applied Analysis, 2011, 10 (1) : 269-286. doi: 10.3934/cpaa.2011.10.269

[2]

Qinqin Zhang. Homoclinic orbits for discrete Hamiltonian systems with local super-quadratic conditions. Communications on Pure & Applied Analysis, 2019, 18 (1) : 425-434. doi: 10.3934/cpaa.2019021

[3]

Oksana Koltsova, Lev Lerman. Hamiltonian dynamics near nontransverse homoclinic orbit to saddle-focus equilibrium. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 883-913. doi: 10.3934/dcds.2009.25.883

[4]

Zheng Yin, Ercai Chen. The conditional variational principle for maps with the pseudo-orbit tracing property. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 463-481. doi: 10.3934/dcds.2019019

[5]

W.-J. Beyn, Y.-K Zou. Discretizations of dynamical systems with a saddle-node homoclinic orbit. Discrete & Continuous Dynamical Systems - A, 1996, 2 (3) : 351-365. doi: 10.3934/dcds.1996.2.351

[6]

S. Secchi, C. A. Stuart. Global bifurcation of homoclinic solutions of Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1493-1518. doi: 10.3934/dcds.2003.9.1493

[7]

Shiwang Ma. Nontrivial periodic solutions for asymptotically linear hamiltonian systems at resonance. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2361-2380. doi: 10.3934/cpaa.2013.12.2361

[8]

Paolo Gidoni, Alessandro Margheri. Lower bound on the number of periodic solutions for asymptotically linear planar Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 585-606. doi: 10.3934/dcds.2019024

[9]

Addolorata Salvatore. Multiple homoclinic orbits for a class of second order perturbed Hamiltonian systems. Conference Publications, 2003, 2003 (Special) : 778-787. doi: 10.3934/proc.2003.2003.778

[10]

Qinqin Zhang. Homoclinic orbits for discrete Hamiltonian systems with indefinite linear part. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1929-1940. doi: 10.3934/cpaa.2015.14.1929

[11]

Juntao Sun, Jifeng Chu, Zhaosheng Feng. Homoclinic orbits for first order periodic Hamiltonian systems with spectrum point zero. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3807-3824. doi: 10.3934/dcds.2013.33.3807

[12]

Cyril Joel Batkam. Homoclinic orbits of first-order superquadratic Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3353-3369. doi: 10.3934/dcds.2014.34.3353

[13]

Zheng Yin, Ercai Chen. Conditional variational principle for the irregular set in some nonuniformly hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6581-6597. doi: 10.3934/dcds.2016085

[14]

Xianfeng Ma, Ercai Chen. Pre-image variational principle for bundle random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 957-972. doi: 10.3934/dcds.2009.23.957

[15]

D. Bartolucci, L. Orsina. Uniformly elliptic Liouville type equations: concentration compactness and a priori estimates. Communications on Pure & Applied Analysis, 2005, 4 (3) : 499-522. doi: 10.3934/cpaa.2005.4.499

[16]

S. Aubry, G. Kopidakis, V. Kadelburg. Variational proof for hard Discrete breathers in some classes of Hamiltonian dynamical systems. Discrete & Continuous Dynamical Systems - B, 2001, 1 (3) : 271-298. doi: 10.3934/dcdsb.2001.1.271

[17]

Ricardo Almeida, Agnieszka B. Malinowska. Fractional variational principle of Herglotz. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2367-2381. doi: 10.3934/dcdsb.2014.19.2367

[18]

Dong-Lun Wu, Chun-Lei Tang, Xing-Ping Wu. Existence and nonuniqueness of homoclinic solutions for second-order Hamiltonian systems with mixed nonlinearities. Communications on Pure & Applied Analysis, 2016, 15 (1) : 57-72. doi: 10.3934/cpaa.2016.15.57

[19]

Amadeu Delshams, Pere Gutiérrez. Exponentially small splitting for whiskered tori in Hamiltonian systems: continuation of transverse homoclinic orbits. Discrete & Continuous Dynamical Systems - A, 2004, 11 (4) : 757-783. doi: 10.3934/dcds.2004.11.757

[20]

Li-Li Wan, Chun-Lei Tang. Existence and multiplicity of homoclinic orbits for second order Hamiltonian systems without (AR) condition. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 255-271. doi: 10.3934/dcdsb.2011.15.255

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (28)
  • HTML views (127)
  • Cited by (0)

Other articles
by authors

[Back to Top]