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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.

MSC:

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
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References:

[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
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