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Existence and nonuniqueness of homoclinic solutions for second-order Hamiltonian systems with mixed nonlinearities
1. | School of Mathematics and Statistics, Southwest University, Chongqing 400715, China |
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. |
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