# zbMATH — the first resource for mathematics

Infinitely many homoclinic solutions for some second order Hamiltonian systems. (English) Zbl 1225.34051
Summary: We investigate the existence of infinitely many homoclinic solutions for a class of second order Hamiltonian systems. Using Zou’s fountain theorem we obtain two new criteria which guarantee that second order Hamiltonian systems have infinitely many homoclinic solutions. Recent results in the literature are generalized and significantly improved.

##### MSC:
 34C37 Homoclinic and heteroclinic solutions to ordinary differential equations 37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
Full Text:
##### References:
 [1] Rabinowitz, P.H., Homoclinic orbits for a class of Hamiltonian systems, Proc. roy. soc. Edinburgh sect. A, 114, 1-2, 33-38, (1990) · Zbl 0705.34054 [2] Coti Zelati, V.; Rabinowitz, P.H., Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potenials, J. amer. math. soc., 4, 693-727, (1991) · Zbl 0744.34045 [3] Ding, Y.; Girardi, M., Periodic and homoclinic solutions to a class of Hamiltonian systems with the potentials changing sign, Dynam. systems appl., 2, 131-145, (1993) · Zbl 0771.34031 [4] Fei, G., The existence of homoclinic orbits for Hamiltonian systems with the potential changing sign, Chinese ann. math. ser. B, 17, 403-410, (1996) · Zbl 0871.58036 [5] Izydorek, M.; Janczewska, J., Homoclinic solutions for a class of second order Hamiltonian systems, J. differ. equ., 219, 2, 375-389, (2005) · Zbl 1080.37067 [6] Lv, X.; Lu, S.; Yan, P., Existence of homoclinic solutions for a class of second order Hamiltonian systems, Nonlinear anal., 72, 390-398, (2010) · Zbl 1186.34059 [7] Korman, P.; Lazer, A.C., Homoclinic orbits for a class of symmetric Hamiltonian systems, Electron. J. differ. equ., 1994, 1-10, (1994) [8] Ou, Z.; Tang, C., Existence of homoclinic solutions for the second order Hamiltonian systems, J. math. anal. appl., 291, 203-213, (2004) · Zbl 1057.34038 [9] Omana, W; Willem, M., Homoclinic orbits for a class of Hamiltonian systems, Differ. int. equ., 5, 5, 1115-1120, (1992) · Zbl 0759.58018 [10] Ding, Y., Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems, Nonlinear anal., 25, 1095-1113, (1995) · Zbl 0840.34044 [11] Zou, W.; Li, S., Infinitely many homoclinic orbits for the second order Hamiltonian systems, Appl. math. lett., 16, 1283-1287, (2003) · Zbl 1039.37044 [12] Yang, J.; Zhang, F., Infinitely many homoclinic orbits for the second order Hamiltonian systems with super-quadratic potentials, Nonlinear anal. RWA, 10, 417-1423, (2009) · Zbl 1162.34328 [13] Zhang, Q.; Liu, C., Infinitely many homoclinic solutions for second order Hamiltonian systems, Nonlinear anal., 72, 894-903, (2010) · Zbl 1178.37063 [14] Wei, J.; Wang, J., Infinitely many homoclinic orbits for second order Hamiltonian systems with general potenial, J. math. anal. appl., 366, 694-699, (2010) · Zbl 1200.37054 [15] Zhang, Z.; Yuan, R., Homoclinic solutions for a class of non-autonomous subquadratic second order Hamiltonian systems, Nonlinear anal., 71, 4125-4130, (2009) · Zbl 1173.34330 [16] Zhang, Z.; Yuan, R., Homoclinic solutions of some second order non-autonomous systems, Nonlinear anal., 71, 5790-5798, (2009) · Zbl 1203.34068 [17] Sun, J.; Chen, H.; Nieto, J.J., Homoclinic solutions for a class of subquadratic second-order Hamiltonian systems, J. math. anal. appl., 373, 1, 20-29, (2011) · Zbl 1230.37079 [18] Zou, W., Variant Fountain theorems and their applications, Manuscripta. math., 104, 343-358, (2001) · Zbl 0976.35026
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.