Periodic solutions of second order non-autonomous Hamiltonian systems with local superquadratic potential. (English) Zbl 1179.34037

The paper deals with the existence of T-periodic solutions for the second order Hamiltonian system
\[ -\ddot{u}(t)=\nabla_uF(t,\,u(t)),\quad a.\;e.\;\;t\in \mathbb{R}. \]
In the case that \(F(t,u)\) is locally superquadratic in \(u\), the authors obtain some existence results for nontrivial T-periodic solutions. The discussion is based on a generalized mountain pass theorem.


34C25 Periodic solutions to ordinary differential equations
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
58E30 Variational principles in infinite-dimensional spaces
Full Text: DOI


[1] Rabinowitz, P.H., On subharmonic solutions of Hamiltonian systems, Comm. pure appl. math., 33, 609-633, (1980) · Zbl 0425.34024
[2] Antonacci, F., Existence of periodic solutions of Hamiltonian systems with potential indefinite in sign, Nonlinear anal., 29, 1353-1364, (1997) · Zbl 0894.34036
[3] Tang, C.L.; Wu, X.P., Periodic solutions of second order Hamiltonian systems with a change sign potential, J. math. anal. appl., 292, 506-516, (2004) · Zbl 1078.34023
[4] Fonda, A.; Ramous, M.; Willem, M., Subharmonic solutions for second order differential equations, Topol. methods nonlinear anal., 1, 49-66, (1993) · Zbl 0803.34029
[5] Long, Y.M., Nonlinear oscillations for classical Hamiltonian systems with bi-even subquadratic potentials, Nonlinear anal., 24, 1665-1671, (1995) · Zbl 0824.34042
[6] Ekeland, I.; Ghoussoub, N., Certain new aspects of the calculus of variations in the large, Bull. amer. math. soc., 39, 207-265, (2002) · Zbl 1064.35054
[7] Li, S.J.; Willem, M., Applications of local linking to critical point theory, J. math. anal. appl., 189, 6-32, (1995) · Zbl 0820.58012
[8] Tao, Z.L.; Tang, C.L., Periodic and subharmonic solutions of second-order Hamiltonian systems, J. math. anal. appl., 293, 435-445, (2004) · Zbl 1042.37047
[9] Fei, G., On periodic solutions of superquadratic Hamiltonian systems, Electron. J. differential equations, 1-12, (2002) · Zbl 0999.37039
[10] Schechter, M., Periodic non-autonomous second order dynamical systems, J. differential equations, 223, 290-302, (2006) · Zbl 1099.34042
[11] Schechter, M., Periodic solution of second order non-autonomous dynamical systems, Boundary value problems, 1-9, (2006)
[12] Rabinowitz, P.H., ()
[13] Ding, Y.H., Variational methods for strongly indefinite problems, (2007), World Scientific Publishing Singapore · Zbl 1133.49001
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.