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Periodic solutions for a class of non-autonomous Hamiltonian systems. (English) Zbl 1071.34039

Summary: We consider the existence of nontrivial periodic solutions for the superlinear Hamiltonian system \[ {\mathcal J}\dot u- A(t)u+\nabla H(t,u)= 0,\quad u\in\mathbb{R}^{2N},\quad t\in\mathbb{R}. \] We prove an abstract result on the existence of a critical point for a real-valued functional on a Hilbert space via a new deformation theorem. Different from the work in the literature, the new deformation theorem is constructed under a Cerami-type condition instead of Palais-Smale-type condition. In addition, the main assumption here is weaker than the usual Ambrosetti-Rabinowitz-type condition \[ 0<\mu H(t, u)\leq u\cdot\nabla H(t,u),\quad \mu> 2,\quad |u|\geq R> 0. \] This result extends theorems given by S. J. Li and M. Willem [J. Math. Anal. Appl. 189, 6–32 (1995; Zbl 0820.58012)] and S. J. Li and A. Szulkin [J. Differ. Equations 112, 226–238 (1994; Zbl 0807.58040)].

MSC:

34C25 Periodic solutions to ordinary differential equations
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
47J30 Variational methods involving nonlinear operators
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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[1] Alama, S.; Li, Y. Y., Existence of solutions for semilinear elliptic equations with indefinite linear part, J. Differential Equations, 96, 89-115 (1992) · Zbl 0766.35009
[2] Bartsch, T.; Ding, Y. H., On a nonlinear Schrödinger equation with periodic potential, Math. Ann., 313, 15-37 (1999) · Zbl 0927.35103
[3] Buffoni, B.; Jeanjean, L.; Stuart, C. A., Existence of nontrivial solutions to a strongly indefinite semilinear equation, Proc. Amer. Math. Soc., 119, 179-186 (1993) · Zbl 0789.35052
[4] Coti-Zelati, V.; Rabinowitz, P., Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potential, J. Amer. Math. Soc., 81, 373-396 (2002)
[5] Y.H. Ding, S.X. Luan, Multiple solutions for a class of nonlinear Schrödinger equations, J. Differential Equations, 207 (2004) 423-457.; Y.H. Ding, S.X. Luan, Multiple solutions for a class of nonlinear Schrödinger equations, J. Differential Equations, 207 (2004) 423-457. · Zbl 1072.35166
[6] Heinz, H. P.; Küpper, T.; Stuart, C. A., Existence and bifurcation of solutions for nonlinear perturbations of the periodic Schrödinger equation, J. Differential Equations, 100, 341-354 (1992) · Zbl 0767.35006
[7] Jeanjean, L., Solutions in spectral gaps for a nonlinear equation of Schrödinger type, J. Differential Equations, 112, 53-80 (1994) · Zbl 0804.35033
[8] Kryszewski, W.; Szulkin, A., Generalized linking theorem with an application to semilinear Schrödinger equations, Adv. Differential Equations, 3, 441-472 (1998) · Zbl 0947.35061
[9] Li, S. J.; Szulkin, A., Period solutions for a class of non-autonomous Hamiltonian systems, J. Differential Equations, 112, 226-238 (1994) · Zbl 0807.58040
[10] Li, S. J.; Willem, M., Applications of local linking to critical point theory, J. Math. Anal. Appl., 189, 6-32 (1995) · Zbl 0820.58012
[11] P. Rabinowitz, Minimax methods in critical point theory with application to differential equations, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 65, 1986.; P. Rabinowitz, Minimax methods in critical point theory with application to differential equations, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 65, 1986. · Zbl 0609.58002
[12] Troestler, C.; Willem, M., Nontrivial solution of a semilinear Schrödinger equation, Commun. Partial Differential Equations, 21, 1431-1449 (1996) · Zbl 0864.35036
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