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Multiple periodic solutions for perturbed relativistic pendulum systems. (English) Zbl 1327.34069

It is proved that the periodically perturbed \(N\)-dimensional relativistic pendulum equation has at least \(N + 1\) geometrically distinct periodic solutions.
Namely, the authors investigate differential systems of the type \[ -(\phi(u'))'= \nabla_u F (t, u) , \quad u(0)-u(T)=0=u'(0)-u'(T), \tag{1} \] where \(\phi\) is defined by \[ \phi(y)=\frac{y}{1-|y|^2}\,. \] Two multiplicity results are obtained: the first one establishes the existence of at least \(N + 1\) geometrically distinct solutions of (1). In the second one, the existence of infinitely many solutions is proved for systems with oscillating potential.
The main idea of the paper is to reduce the singular problem (1) to an equivalent non-singular one to which classical variational methods can be applied.

MSC:

34C25 Periodic solutions to ordinary differential equations
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
34B16 Singular nonlinear boundary value problems for ordinary differential equations
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[1] Bereanu, Cristian; Jebelean, Petru, Multiple critical points for a class of periodic lower semicontinuous functionals, Discrete Contin. Dyn. Syst., 33, 1, 47-66 (2013) · Zbl 1281.34024 · doi:10.3934/dcds.2013.33.47
[2] Coelho, Isabel; Corsato, Chiara; Obersnel, Franco; Omari, Pierpaolo, Positive solutions of the Dirichlet problem for the one-dimensional Minkowski-curvature equation, Adv. Nonlinear Stud., 12, 3, 621-638 (2012) · Zbl 1263.34028
[3] [CCR] I. Coelho, C. Corsato and S. Rivetti, Positive radial solutions of the Dirichlet problem for the Minkowski-curvature equation in a ball, Topological Meth. Nonlinear Anal., to appear. · Zbl 1366.35029
[4] Habets, Patrick; Man{\'a}sevich, Ra{\'u}l; Zanolin, Fabio, A nonlinear boundary value problem with potential oscillating around the first eigenvalue, J. Differential Equations, 117, 2, 428-445 (1995) · Zbl 0821.34017 · doi:10.1006/jdeq.1995.1060
[5] Habets, Patrick; Serra, Enrico; Tarallo, Massimo, Multiplicity results for boundary value problems with potentials oscillating around resonance, J. Differential Equations, 138, 1, 133-156 (1997) · Zbl 0887.35059 · doi:10.1006/jdeq.1997.3267
[6] L{\`“u}, Haishen; O”Regan, Donal; Agarwal, Ravi P., On the existence of multiple periodic solutions for the vector \(p\)-Laplacian via critical point theory, Appl. Math., 50, 6, 555-568 (2005) · Zbl 1099.34021 · doi:10.1007/s10492-005-0037-8
[7] Mawhin, Jean, Multiplicity of solutions of variational systems involving \(\phi \)-Laplacians with singular \(\phi\) and periodic nonlinearities, Discrete Contin. Dyn. Syst., 32, 11, 4015-4026 (2012) · Zbl 1260.34076 · doi:10.3934/dcds.2012.32.4015
[8] Mawhin, Jean; Willem, Michel, Critical point theory and Hamiltonian systems, Applied Mathematical Sciences 74, xiv+277 pp. (1989), Springer-Verlag, New York · Zbl 0676.58017 · doi:10.1007/978-1-4757-2061-7
[9] Obersnel, Franco; Omari, Pierpaolo, The periodic problem for curvature-like equations with asymmetric perturbations, J. Differential Equations, 251, 7, 1923-1971 (2011) · Zbl 1241.34051 · doi:10.1016/j.jde.2011.06.014
[10] Rabinowitz, Paul H., On a class of functionals invariant under a \({\bf Z}^n\) action, Trans. Amer. Math. Soc., 310, 1, 303-311 (1988) · Zbl 0718.34057 · doi:10.2307/2001123
[11] Zhang, Peng; Tang, Chun-Lei, Infinitely many periodic solutions for nonautonomous sublinear second-order Hamiltonian systems, Abstr. Appl. Anal., 2010, Art. ID 620438, 10 pp. pp. · Zbl 1203.37104 · doi:10.1155/2010/620438
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