×

Periodic solutions of singular radially symmetric systems with superlinear growth. (English) Zbl 1251.34056

The authors prove the existence of infinitely many periodic solutions of a forced radially symmetric systems of second-order ODEs. The nonlinearity exhibits a singularity of repulsive type at the origin and has superlinear growth at infinity. The obtained solutions have periods, which are large integer multiples of the period of the forcing term. They rotate exactly once around the origin in their period time and have a fast oscillating radial component. The rather delicate proof is based on topological degree arguments.

MSC:

34C25 Periodic solutions to ordinary differential equations
47H11 Degree theory for nonlinear operators
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Ambrosetti, A.; Coti Zelati, V., Periodic Solutions of Singular Lagrangian Systems, Progress in Nonlinear Differential Equations and their Applications, vol 10 (1993), Boston: Birkhäuser, Boston · Zbl 0785.34032
[2] Arnold, V. I., Mathematical Methods of Classical Mechanics, Grad. Texts in Math., vol. 60 (1978), Heidelberg: Springer, Heidelberg · Zbl 0386.70001
[3] Benci, V., Normal modes of a Lagrangian system constrained in a potential well, Ann. Inst. H. Poincaré, 1, 379-400 (1984) · Zbl 0561.58006
[4] Capietto, A.; Henrard, M.; Mawhin, J.; Zanolin, F., A continuation approach to some forced superlinear Sturm-Liouville boundary value problems, Topol. Meth. Nonlin. Anal., 3, 81-100 (1994) · Zbl 0808.34028
[5] Capietto, A.; Mawhin, J.; Zanolin, F., A continuation approach to superlinear periodic boundary value problems, J. Differ. Equ., 88, 347-395 (1990) · Zbl 0718.34053 · doi:10.1016/0022-0396(90)90102-U
[6] Capietto, A.; Mawhin, J.; Zanolin, F., A duality theorem for coincidence equations with applications to superlinear Neumann boundary value problems, J. Niger. Math. Soc., 11, 83-105 (1992)
[7] Capietto, A.; Mawhin, J.; Zanolin, F., On the existence of two solutions with a prescribed number of zeros for a superlinear two-point boundary value problem, Topol. Meth. Nonlin. Anal., 6, 175-188 (1995) · Zbl 0849.34018
[8] Capozzi, A.; Greco, C.; Salvatore, A., Lagrangian systems in the presence of singularities, Proc. Amer. Math. Soc., 102, 125130 (1988) · Zbl 0664.34054 · doi:10.1090/S0002-9939-1988-0915729-0
[9] Chu, J.; Franco, D., Non-collision periodic solutions of second order singular dynamical systems, J. Math. Anal. Appl., 344, 898-905 (2008) · Zbl 1153.34335 · doi:10.1016/j.jmaa.2008.03.041
[10] Coti Zelati, V., Dynamical systems with effective-like potential, Nonlinear Anal., 12, 209-222 (1988) · Zbl 0648.34050 · doi:10.1016/0362-546X(88)90035-1
[11] Coti Zelati, V., Li, S., Wu, S.: Periodic solutions for a class of singular nonautonomous second order systems in a potential well. In: Variational Methods in Nonlinear Analysis. Gordon and Breach, Basel (1995) · Zbl 0847.34041
[12] Degiovanni, M., Giannoni, F., Marino, A.: Dynamical systems with Newtonian type potentials, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 81 (1987) (n.3, 271-277, 1988) · Zbl 0667.70010
[13] Del Pino, M.; Manásevich, R., Infinitely many T-periodic solutions for a problem arising in nonlinear elasticity, J. Differ. Equ., 103, 260-277 (1993) · Zbl 0781.34032 · doi:10.1006/jdeq.1993.1050
[14] Ding, T.; Zanolin, F., Periodic solutions of Duffing’s equations with superquadratic potential, J. Differ. Equ., 97, 328-378 (1992) · Zbl 0763.34030 · doi:10.1016/0022-0396(92)90076-Y
[15] Ding, Y.; Li, S., Periodic solutions of some singular dynamical systems in a potential well, Chin. J. Contemp. Math., 13, 1992, 299-307 (1993)
[16] Fonda, A., Periodic solutions for a conservative system of differential equations with a singularity of repulsive type, Nonlinear Anal., 24, 667-676 (1995) · Zbl 0830.34033 · doi:10.1016/0362-546X(94)00118-2
[17] Fonda, A.; Manásevich, R.; Zanolin, F., Subharmonic solutions for some second-order differential equations with singularities, SIAM J. Math. Anal., 24, 1294-1311 (1993) · Zbl 0787.34035 · doi:10.1137/0524074
[18] Fonda, A.; Toader, R., Periodic orbits of radially symmetric Keplerian-like systems: a topological degree approach, J. Differ. Equ., 244, 3235-3264 (2008) · Zbl 1168.34031 · doi:10.1016/j.jde.2007.11.005
[19] Fonda, A., Toader, R.: Periodic orbits of radially symmetric systems with a singularity: the repulsive case. Adv. Nonlinear Stud. (to appear) · Zbl 1241.34048
[20] Fonda, A., Toader, R.: Periodic solutions of radially symmetric perturbations of Newtonian systems, preprint. Available on-line at www.dmi.units.it/pubblicazioni/Quaderni_Matematici/598_2009.pdf · Zbl 1248.34051
[21] Fonda, A., Toader, R.: Radially symmetric systems with a singularity and asymptotically linear growth. Nonlinear Anal. (to appear) · Zbl 1218.34047
[22] Fonda, A.; Ureña, A., Periodic, subharmonic, and quasi-periodic oscillations under the action of a central force, Discret. Cont. Dynam. Syst., 29, 169-192 (2011) · Zbl 1231.34073 · doi:10.3934/dcds.2011.29.169
[23] García-Huidobro, M.; Manásevich, R.; Zanolin, F., Strongly nonlinear second-order ODEs with rapidly growing terms, J. Math. Anal. Appl., 202, 1-26 (1996) · Zbl 0991.34007 · doi:10.1006/jmaa.1996.0300
[24] Gaines, E., Mawhin, J.: Coincidence Degree and Nonlinear Differential Equations. Lecture Notes in Math, vol. 568. Springer, Berlin (1977) · Zbl 0339.47031
[25] Gordon, W. B., Conservative dynamical systems involving strong forces, Trans. Am. Math. Soc., 204, 113-135 (1975) · Zbl 0276.58005 · doi:10.1090/S0002-9947-1975-0377983-1
[26] Habets, P.; Sanchez, L., Periodic solutions of dissipative dynamical systems with singular potentials, Differ. Integr. Equ., 3, 1139-1149 (1990) · Zbl 0724.34049
[27] Jacobowitz, H., Periodic solutions of x′′ + f(x, t) = 0 via the Poincaré-Birkhoff theorem, J. Differ. Equ., 20, 37-52 (1976) · Zbl 0285.34028 · doi:10.1016/0022-0396(76)90094-2
[28] Krasnosel’Skii, M. A.; Zabreiko, P. P., Geometrical Methods of Nonlinear Analysis (1984), Berlin: Springer, Berlin · Zbl 0546.47030
[29] Lazer, A. C.; Solimini, S., On periodic solutions of nonlinear differential equations with singularities, Proc. Am. Math. Soc., 99, 109-114 (1987) · Zbl 0616.34033 · doi:10.1090/S0002-9939-1987-0866438-7
[30] Levi, M., Quasiperiodic motions in superquadratic time-periodic potentials, Comm. Math. Phys., 143, 43-83 (1991) · Zbl 0744.34043 · doi:10.1007/BF02100285
[31] Llibre, J.; Ortega, R., On the families of periodic orbits of the Sitnikov problem, SIAM J. Appl. Dyn. Syst., 7, 561-576 (2008) · Zbl 1159.70010 · doi:10.1137/070695253
[32] Mawhin, J., Leray-Schauder degree: a half century of extensions and applications, Topol. Meth. Nonlin. Anal., 14, 195-228 (1999) · Zbl 0957.47045
[33] Morris, G. R., An infinite class of periodic solutions of ẍ + 2x^3 = p(t), Proc. Cambridge Philos. Soc., 61, 157-164 (1965) · Zbl 0134.07203 · doi:10.1017/S0305004100038743
[34] Van Noort, M.; Porter, M. A.; Yi, Y.; Chow, S.-N., Quasiperiodic dynamics in Bose-Einstein condensates in periodic lattices and superlattices, J. Nonlinear Sci., 17, 59-83 (2007) · Zbl 1114.37048 · doi:10.1007/s00332-005-0723-4
[35] Rabinowitz, P. H., Some aspects of nonlinear eigenvalue problems, Rocky Mount. J. Math., 3, 161-202 (1973) · Zbl 0255.47069 · doi:10.1216/RMJ-1973-3-2-161
[36] Rabinowitz, P.H.: Variational methods for Hamiltonian systems. In: Handbook of Dynamical Systems, vol. 1A pp. 1091-1127. North-Holland, Amsterdam (2002) · Zbl 1048.37055
[37] Serra, E.; Terracini, S., Noncollision solutions to some singular minimization problems with Keplerian-like potentials, Nonlinear Anal., 22, 45-62 (1994) · Zbl 0813.70006 · doi:10.1016/0362-546X(94)90004-3
[38] Solimini, S., On forced dynamical systems with a singularity of repulsive type, Nonlinear Anal., 14, 489-500 (1990) · Zbl 0708.34041 · doi:10.1016/0362-546X(90)90037-H
[39] Torres, P. J., Non-collision periodic solutions of forced dynamical systems with weak singularities, Discret. Contin. Dyn. Syst., 11, 693-698 (2004) · Zbl 1063.34035 · doi:10.3934/dcds.2004.11.693
[40] Zeidler, E., Nonlinear Functional Analysis and its Applications, vol. 1 (1986), New York-Heidelberg: Springer, New York-Heidelberg · Zbl 0583.47050
[41] Zhong-Heng, G.; Solecki, R., Free and forced finite amplitude oscillations of an elastic thick walled hollow sphere made in incompressible material, Arch. Mech. Stos., 25, 427-433 (1963) · Zbl 0151.39003
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.