American Institute of Mathematical Sciences

Advanced
January  2016, 15(1): 57-72. doi: 10.3934/cpaa.2016.15.57

Existence and nonuniqueness of homoclinic solutions for second-order Hamiltonian systems with mixed nonlinearities

Dong-Lun Wu 1, , Chun-Lei Tang 1, and  Xing-Ping Wu 1,

1. 

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

Received  December 2014 Revised  October 2015 Published  December 2015

In this paper, we study the existence of homoclinic solutions to the following second-order Hamiltonian systems \begin{eqnarray} \ddot{u}(t)-L(t)u(t)+\nabla W(t,u(t))=0,\quad \forall t\in R, \end{eqnarray} where $L(t)$ is a symmetric and positive definite matrix for all $t\in R$. The nonlinear potential $W$ is a combination of superlinear and sublinear terms. By different conditions on the superlinear and sublinear terms, we obtain existence and nonuniqueness of nontrivial homoclinic solutions to above systems.
Keywords: indefinite signs., Multiple homoclinic solutions, second-order Hamiltonian systems, mixed nonlinearities, variational methods.
Mathematics Subject Classification: Primary: 34C37, 37J45; Secondary: 47J3.
Citation: 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 and Applied Analysis, 2016, 15 (1) : 57-72. doi: 10.3934/cpaa.2016.15.57
References:
[1]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.

[2]

A. Ambrosetti and V. Coti Zelati, Multiple homoclinic orbits for a class of conservative systems, Rend. Sem. Mat. Univ. Padova, 89 (1993), 177-194.

[3]

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

[4]

P. C. Carrião 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.

[5]

H. W. Chen and Z. M. He, Infinitely many homoclinic solutions for a class of second-order Hamiltonian systems, Advances in Difference Equations, 2014 (2014):161.

[6]

Y. H. 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.

[7]

Y. H. Ding and S. J. Li, Homoclinic orbits for first order Hamiltonian systems, J. Math. Anal. Appl., 189 (1995), 585-601. doi: 10.1006/jmaa.1995.1037.

[8]

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

[9]

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

[10]

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

[11]

S. P. Lu, Homoclinic solutions for a nonlinear second order differential system with p-Laplacian operator, Nonlinear Anal. RWA., 12 (2011), 525-534. doi: 10.1016/j.nonrwa.2010.06.037.

[12]

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.

[13]

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

[14]

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.

[15]

E. Paturel, Multiple homoclinic orbits for a class of Hamiltonian systems, Calc. Var. Partial Differential Equations, 12 (2001), 117-143. doi: 10.1007/PL00009909.

[16]

P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, in CBMS, Regional Conf. Ser. in Math., vol. 65, Amer. Math. Soc., Providence, RI, 1986.

[17]

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.

[18]

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.

[19]

J. Sun, H. Chen and J. 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.

[20]

X. H. Tang and X. Y. Lin, Existence of infinitely many homoclinic orbits in Hamiltonian systems, Pro. Roy. Soc. Edin., 141 (2011), 1103-1119. doi: 10.1017/S0308210509001346.

[21]

X. H. Tang and X. Y. Lin, Infinitely many homoclinic orbits for Hamiltonian systems with indefinite sign subquadratic potentials, Nonlinear Anal., 74 (2011), 6314-6325. doi: 10.1016/j.na.2011.06.010.

[22]

L.-L. Wan and C.-L. Tang, Existence of homoclinic orbits for second order Hamiltonian systems without (AR) condition, Nonlinear Anal., 74 (2011), 5303-5313. doi: 10.1016/j.na.2011.05.011.

[23]

L. Yang, H. Chen and J. Sun, Infinitely many homoclinic solutions for some second order Hamiltonian systems, Nonlinear Anal., 74 (2011), 6459-6468. doi: 10.1016/j.na.2011.06.029.

[24]

Y. W. Y and C.-L. Tang, Multiple homoclinic solutions for second-order perturbed Hamiltonian systems, Studies in Applied Mathematics, 132 (2014), 112-137. doi: 10.1111/sapm.12023.

[25]

R. Yuan and Z. Zhang, Homoclinic solutions for a class of second order Hamiltonian systems, Results in Math., 61 (2012) 195-208. doi: 10.1007/s00025-010-0088-3.

[26]

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

[27]

Z. Zhang, X. Tian and R. Yuan, Homoclinic solutions for subquadratic Hamiltonian systems without coercive conditions, Taiwanese J. Math., 18 (2014), 1089-1105. doi: 10.11650/tjm.18.2014.3508.

show all references

References:
[1]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.

[2]

A. Ambrosetti and V. Coti Zelati, Multiple homoclinic orbits for a class of conservative systems, Rend. Sem. Mat. Univ. Padova, 89 (1993), 177-194.

[3]

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

[4]

P. C. Carrião 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.

[5]

H. W. Chen and Z. M. He, Infinitely many homoclinic solutions for a class of second-order Hamiltonian systems, Advances in Difference Equations, 2014 (2014):161.

[6]

Y. H. 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.

[7]

Y. H. Ding and S. J. Li, Homoclinic orbits for first order Hamiltonian systems, J. Math. Anal. Appl., 189 (1995), 585-601. doi: 10.1006/jmaa.1995.1037.

[8]

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

[9]

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

[10]

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

[11]

S. P. Lu, Homoclinic solutions for a nonlinear second order differential system with p-Laplacian operator, Nonlinear Anal. RWA., 12 (2011), 525-534. doi: 10.1016/j.nonrwa.2010.06.037.

[12]

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.

[13]

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

[14]

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.

[15]

E. Paturel, Multiple homoclinic orbits for a class of Hamiltonian systems, Calc. Var. Partial Differential Equations, 12 (2001), 117-143. doi: 10.1007/PL00009909.

[16]

P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, in CBMS, Regional Conf. Ser. in Math., vol. 65, Amer. Math. Soc., Providence, RI, 1986.

[17]

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.

[18]

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.

[19]

J. Sun, H. Chen and J. 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.

[20]

X. H. Tang and X. Y. Lin, Existence of infinitely many homoclinic orbits in Hamiltonian systems, Pro. Roy. Soc. Edin., 141 (2011), 1103-1119. doi: 10.1017/S0308210509001346.

[21]

X. H. Tang and X. Y. Lin, Infinitely many homoclinic orbits for Hamiltonian systems with indefinite sign subquadratic potentials, Nonlinear Anal., 74 (2011), 6314-6325. doi: 10.1016/j.na.2011.06.010.

[22]

L.-L. Wan and C.-L. Tang, Existence of homoclinic orbits for second order Hamiltonian systems without (AR) condition, Nonlinear Anal., 74 (2011), 5303-5313. doi: 10.1016/j.na.2011.05.011.

[23]

L. Yang, H. Chen and J. Sun, Infinitely many homoclinic solutions for some second order Hamiltonian systems, Nonlinear Anal., 74 (2011), 6459-6468. doi: 10.1016/j.na.2011.06.029.

[24]

Y. W. Y and C.-L. Tang, Multiple homoclinic solutions for second-order perturbed Hamiltonian systems, Studies in Applied Mathematics, 132 (2014), 112-137. doi: 10.1111/sapm.12023.

[25]

R. Yuan and Z. Zhang, Homoclinic solutions for a class of second order Hamiltonian systems, Results in Math., 61 (2012) 195-208. doi: 10.1007/s00025-010-0088-3.

[26]

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

[27]

Z. Zhang, X. Tian and R. Yuan, Homoclinic solutions for subquadratic Hamiltonian systems without coercive conditions, Taiwanese J. Math., 18 (2014), 1089-1105. doi: 10.11650/tjm.18.2014.3508.

[1]

Xiaoping Wang. Ground state homoclinic solutions for a second-order Hamiltonian system. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 2163-2175. doi: 10.3934/dcdss.2019139
[2]

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
[3]

Ying Lv, Yan-Fang Xue, Chun-Lei Tang. Ground state homoclinic orbits for a class of asymptotically periodic second-order Hamiltonian systems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1627-1652. doi: 10.3934/dcdsb.2020176
[4]

Yan Liu, Fei Guo. Multiplicity of periodic solutions for second-order perturbed Hamiltonian systems with local superquadratic conditions. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2022098
[5]

Qiong Meng, X. H. Tang. Solutions of a second-order Hamiltonian system with periodic boundary conditions. Communications on Pure and Applied Analysis, 2010, 9 (4) : 1053-1067. doi: 10.3934/cpaa.2010.9.1053
[6]

C. Rebelo. Multiple periodic solutions of second order equations with asymmetric nonlinearities. Discrete and Continuous Dynamical Systems, 1997, 3 (1) : 25-34. doi: 10.3934/dcds.1997.3.25
[7]

Norimichi Hirano, Zhi-Qiang Wang. Subharmonic solutions for second order Hamiltonian systems. Discrete and Continuous Dynamical Systems, 1998, 4 (3) : 467-474. doi: 10.3934/dcds.1998.4.467
[8]

Marc Henrard. Homoclinic and multibump solutions for perturbed second order systems using topological degree. Discrete and Continuous Dynamical Systems, 1999, 5 (4) : 765-782. doi: 10.3934/dcds.1999.5.765
[9]

Li-Li Wan, Chun-Lei Tang. Existence and multiplicity of homoclinic orbits for second order Hamiltonian systems without (AR) condition. Discrete and Continuous Dynamical Systems - B, 2011, 15 (1) : 255-271. doi: 10.3934/dcdsb.2011.15.255
[10]

Qinqin Zhang. Homoclinic orbits for discrete Hamiltonian systems with indefinite linear part. Communications on Pure and Applied Analysis, 2015, 14 (5) : 1929-1940. doi: 10.3934/cpaa.2015.14.1929
[11]

Qiong Meng, X. H. Tang. Multiple solutions of second-order ordinary differential equation via Morse theory. Communications on Pure and Applied Analysis, 2012, 11 (3) : 945-958. doi: 10.3934/cpaa.2012.11.945
[12]

Juhong Kuang, Weiyi Chen, Zhiming Guo. Periodic solutions with prescribed minimal period for second order even Hamiltonian systems. Communications on Pure and Applied Analysis, 2022, 21 (1) : 47-59. doi: 10.3934/cpaa.2021166
[13]

Xiangjin Xu. Multiple solutions of super-quadratic second order dynamical systems. Conference Publications, 2003, 2003 (Special) : 926-934. doi: 10.3934/proc.2003.2003.926
[14]

Leonardo Colombo, David Martín de Diego. Second-order variational problems on Lie groupoids and optimal control applications. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6023-6064. doi: 10.3934/dcds.2016064
[15]

Jae-Hong Pyo, Jie Shen. Normal mode analysis of second-order projection methods for incompressible flows. Discrete and Continuous Dynamical Systems - B, 2005, 5 (3) : 817-840. doi: 10.3934/dcdsb.2005.5.817
[16]

S. Secchi, C. A. Stuart. Global bifurcation of homoclinic solutions of Hamiltonian systems. Discrete and Continuous Dynamical Systems, 2003, 9 (6) : 1493-1518. doi: 10.3934/dcds.2003.9.1493
[17]

Maurizio Grasselli, Morgan Pierre. Convergence to equilibrium of solutions of the backward Euler scheme for asymptotically autonomous second-order gradient-like systems. Communications on Pure and Applied Analysis, 2012, 11 (6) : 2393-2416. doi: 10.3934/cpaa.2012.11.2393
[18]

Juntao Sun, Jifeng Chu, Zhaosheng Feng. Homoclinic orbits for first order periodic Hamiltonian systems with spectrum point zero. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3807-3824. doi: 10.3934/dcds.2013.33.3807
[19]

Cyril Joel Batkam. Homoclinic orbits of first-order superquadratic Hamiltonian systems. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3353-3369. doi: 10.3934/dcds.2014.34.3353
[20]

Xingyong Zhang, Xianhua Tang. Some united existence results of periodic solutions for non-quadratic second order Hamiltonian systems. Communications on Pure and Applied Analysis, 2014, 13 (1) : 75-95. doi: 10.3934/cpaa.2014.13.75

2021 Impact Factor: 1.273

Tools

Metrics

  • PDF downloads (60)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy