## On superquadratic periodic systems with indefinite linear part.(English)Zbl 1198.34066

The authors consider the second order system of ordinary differential equations
$-x''-A(t)x = \nabla F(t,x),\quad x\in\mathbb{R}^N,$
where $$A$$ is an $$N\times N$$ symmetric matrix with periodic entries, $$F$$ is periodic in $$t$$ and $$\nabla F(t,0)=0$$. They show that, under a rather weak superquadraticity condition on $$F$$ at infinity, this system has a nontrivial periodic solution. The proof uses Morse theory as follows. The superquadraticity condition implies that the Euler-Lagrange functional $$\varphi$$ corresponding to the problem satisfies the Cerami condition (hence, the usual deformation lemma holds) and the critical groups at infinity $$C_k(\varphi,\infty)=0$$ for all $$k$$. The conditions on $$F$$ at 0 imply that the critical group $$C_d(\varphi,0)\neq 0$$ for a certain $$d$$. Hence, there must exist a critical point $$x_0\neq 0$$ for $$\varphi$$ (it is also shown that $$\varphi(x_0)\neq 0$$).

### MSC:

 34C25 Periodic solutions to ordinary differential equations 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
Full Text:

### References:

 [1] Rabinowitz, P., Periodic solutions of Hamiltonian systems, Comm. pure appl. math., 31, 157-184, (1978) · Zbl 0358.70014 [2] Ekeland, I.; Ghoussoub, N., Selected new aspects of the calculus of variations in the large, Bull. amer. math. soc., 39, 207-265, (2002) · Zbl 1064.35054 [3] Li, S.; Willem, M., Applications of local linking to critical point theory, J. math. anal. appl., 189, 6-32, (1995) · Zbl 0820.58012 [4] Long, Y., Multiple solutions of perturbed superquadratic second order Hamiltonian systems, Trans. amer. math. soc., 311, 749-780, (1989) · Zbl 0676.34026 [5] Xu, X., Periodic solutions for nonautonomous Hamiltonian systems possessing superquadratic potentials, Nonlinear anal., 51, 941-955, (2002) · Zbl 1215.37041 [6] Fei, G., On periodic solutions of superquadratic Hamiltonian systems, Electron. J. differential equations, 8, 12, (2002) · Zbl 0999.37039 [7] Gasiński, L.; Papageorgiou, N.S., Nonlinear analysis, (2006), Chapman & Hall/CRC Press Boca Raton, Fl · Zbl 1027.35041 [8] He, X.; Wu, X., Periodic solutions for a class of nonautonomous second order Hamiltonian systems, J. math. anal. appl., 341, 1354-1364, (2008) · Zbl 1133.37025 [9] Luan, S.; Mao, A., Periodic solutions for a class of nonautonomous Hamiltonian systems, Nonlinear anal., 61, 1413-1426, (2005) · Zbl 1071.34039 [10] Tao, Z.; Yao, S.; Wu, S., Periodic solutions for a class of superquadratic Hamiltonian systems, J. math. anal. appl., 331, 152-158, (2007) · Zbl 1123.34311 [11] Barletta, G.; Papageorgiou, N.S., Nonautonomous second order periodic systems:existence and multiplicity of solutions, J. nonlinear convex anal., 8, 373-390, (2007) · Zbl 1158.34030 [12] Motreanu, D.; Motreanu, V.; Papageorgiou, N.S., Periodic solutions for nonautonomous systems with nonsmooth quadratic or superquadratic potential, Topol. methods nonlinear anal., 24, 269-296, (2004) · Zbl 1070.34068 [13] Motreanu, D.; Motreanu, V.; Papageorgiou, N.S., Two nontrivial solutions for periodic systems with indefinite linear part, Discrete contin. dyn. syst. A, 19, 197-210, (2007) · Zbl 1146.34011 [14] Kandilakis, D.; Kourogenis, N.; Papageorgiou, N.S., Two nontrivial critical points for nonsmooth functionals via local linking and applications, J. global optim., 34, 219-244, (2006) · Zbl 1105.49019 [15] Gasiński, L.; Papageorgiou, N.S., Nonsmooth critical point theory and nonlinear boundary value problems, (2005), Chapman & Hall/CRC Press Boca Raton, Fl · Zbl 1058.58005 [16] Perera, K., Critical groups of critical points produced by local linking with applications, Abstr. appl. anal., 3, 437-446, (1998) · Zbl 0972.58008 [17] Showalter, R., Hilbert space methods for partial differential equations, (1977), Pitman London · Zbl 0364.35001 [18] Mawhin, J.; Willem, M., () [19] Chang, K.-C., Infinite dimensional Morse theory and multiple solution problems, (1993), Birkhäuser Boston [20] Bartolo, P.; Benci, V.; Fortunato, D., Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity, Nonlinear anal., 7, 981-1012, (1983) · Zbl 0522.58012 [21] Bartsch, T.; Li, S., Critical point theory for asymptotically quadratic functionals and applications to problems with resonance, Nonlinear anal., 28, 419-441, (1997) · Zbl 0872.58018 [22] Dugundji, J., Topology, (1966), Allyn and Bacon Boston · Zbl 0144.21501 [23] Granas, D.; Dugundji, J., Fixed point theory, (2003), Springer-Verlag New York · Zbl 1025.47002 [24] Su, J., Multiplicity results for asymptotically linear elliptic problems at resonance, J. math. anal. appl., 278, 397-408, (2003) · Zbl 1290.35109
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.