Tang, Chunlei; Wu, Xingping Periodic solutions for second order systems with not uniformly coercive potential. (English) Zbl 0999.34039 J. Math. Anal. Appl. 259, No. 2, 386-397 (2001). The authors study the second-order system \[ \ddot u(t)=\nabla F(t,u(t))\quad \text{a.e.}\quad t\in[0, T],\qquad u(0)-u(T)=\dot u(0)-\dot u(T)=0, \] with locally coercive potential, that is \(F(t,x)\rightarrow\infty\) a.e. for \(t\) in some positive measure subset of \([0, T]\). Existence and multiplicity of periodic solutions are obtained. The result is established using an analogy of Egorov’s theorem, properties of subadditive functions, the least action principle, and a three-critical-point theorem proposed by Brezis and Nirenberg. Reviewer: Ivan Ginchev (Varna) Cited in 1 ReviewCited in 80 Documents MSC: 34C25 Periodic solutions to ordinary differential equations Keywords:periodic solutions; second-order systems; subadditivity; coercivity; Sobolev’s inequality; critical points × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Mawhin, J.; Willem, M., Critical Point Theory and Hamiltonian Systems (1989), Springer-Verlag: Springer-Verlag New York · Zbl 0676.58017 [2] Berger, M. S.; Schechter, M., On the solvability of semilinear gradient operator equations, Adv. Math., 25, 97-132 (1977) · Zbl 0354.47025 [3] Willem, M., Oscillations forcées de systèmes hamiltoniens, Public, Sémin. Analyse Non Linéaire (1981), Univ. Besancon · Zbl 0482.70020 [4] Mawhin, J., Semi-coercive monotone variational problems, Acad. Roy. Belg. Bull. Cl. Sci., 73, 118-130 (1987) · Zbl 0647.49007 [5] Tang, C. L., Periodic solutions of nonautonomous second order systems with γ-quasisubadditive potential, J. Math. Anal. Appl., 189, 671-675 (1995) · Zbl 0824.34043 [6] Tang, C. L., Periodic solutions of nonautonomous second order systems with sublinear nonlinearity, Proc. Amer. Math. Soc., 126, 3263-3270 (1998) · Zbl 0902.34036 [7] Long, Y. M., Nonlinear oscillations for classical Hamiltonian systems with bieven subquadratic potentials, Nonlinear Anal., 24, 1665-1671 (1995) · Zbl 0824.34042 [8] Tang, C. L., Periodic solutions of nonautonomous second order systems, J. Math. Anal. Appl., 202, 465-469 (1996) · Zbl 0857.34044 [9] Brezis, H.; Nirenberg, L., Remarks on finding critical points, Comm. Pure Appl. Math., 44, 939-963 (1991) · Zbl 0751.58006 [10] Tang, C. L., Existence and multiplicity of periodic solutions of nonautonomous second order systems, Nonlinear Anal., 32, 299-304 (1998) · Zbl 0949.34032 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.