# American Institute of Mathematical Sciences

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

 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.
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 & Applied Analysis, 2016, 15 (1) : 57-72. doi: 10.3934/cpaa.2016.15.57
##### References:
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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 [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] P. Korman and A. C. Lazer, Homoclinic orbits for a class of symmetric Hamiltonian systems, Electron. J. Differential Equations, 1994 (1994), 1-10.  Google Scholar [11] S. P. 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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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [21] X. H. Tang and X. Y. 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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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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##### References:
 [1] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.  Google Scholar [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.  Google Scholar [3] K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems, Birkhäuser, Boston 1993. doi: 10.1007/978-1-4612-0385-8.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] P. Korman and A. C. Lazer, Homoclinic orbits for a class of symmetric Hamiltonian systems, Electron. J. Differential Equations, 1994 (1994), 1-10.  Google Scholar [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.  Google Scholar [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.  Google Scholar [13] W. Omana and M. Willem, Homoclinic orbits for a class of Hamiltonian systems, Differential and Integral Equations, 5 (1992), 1115-1120.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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