In this paper, by using the least action principle, Sobolev's inequality and Wirtinger's inequality, some existence theorems are obtained for periodic solutions of second-order Hamiltonian systems with a p-Laplacian under subconvex condition, sublinear growth condition and linear growth condition. Our results generalize and improve those in the literature.
@article{bwmeta1.element.doi-10_2478_v10062-010-0008-8,
author = {Xianhua Tang and Xingyong Zhang},
title = {
Periodic solutions for second-order Hamiltonian systems with a
p
-Laplacian
},
journal = {Annales UMCS, Mathematica},
volume = {64},
year = {2010},
pages = {93-113},
zbl = {1219.34057},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_v10062-010-0008-8}
}
Xianhua Tang; Xingyong Zhang.
Periodic solutions for second-order Hamiltonian systems with a
p
-Laplacian
. Annales UMCS, Mathematica, Tome 64 (2010) pp. 93-113. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_v10062-010-0008-8/
Berger, M. S., Schechter, M., On the solvability of semilinear gradient operator equations, Adv. Math. 25 (1977), 97-132. | Zbl 0354.47025
Mawhin, J., Semi-coercive monotone variational problems, Acad. Roy. Belg. Bull. Cl. Sci. 73 (1987), 118-130. | Zbl 0647.49007
Mawhin, J., Willem, M., Critical Point Theory and Hamiltonian Systems, Springer-Verlag, New York, 1989. | Zbl 0676.58017
Mawhin, J., Willem, M., Critical points of convex perturbations of some indefinite quadratic forms and semilinear boundary value problems at resonance, Ann. Inst. H. Poincaré Anal. Non Linéire 3 (1986), 431-453. | Zbl 0678.35091
Rabinowitz, P. H., On subharmonic solutions of Hamiltonian systems, Comm. Pure Appl. Math. 33 (1980), 609-633. | Zbl 0425.34024
Rabinowitz, P. H., Minimax methods in critical point theory with applications to differential equations, in: CBMS Regional Conf. Ser. in Math., Vol. 65, American Mathematical Society, Providence, RI, 1986.
Tang, C. L., Periodic solutions of nonautonomous second order systems with γ-quasisubadditive potential, J. Math. Anal. Appl. 189 (1995), 671-675. | Zbl 0824.34043
Tang, C. L., Periodic solutions of nonautonomous second order systems, J. Math. Anal. Appl. 202 (1996), 465-469. | Zbl 0857.34044
Tang, C. L., Periodic solutions of nonautonomous second order systems with sublinear nonlinearity, Proc. Amer. Math. Soc. 126 (1998), 3263-3270. | Zbl 0902.34036
Tang, C. L., Wu, X. P., Periodic solutions for second order systems with not uniformly coercive potential, J. Math. Anal. Appl. 259 (2001), 386-397. | Zbl 0999.34039
Willem, M., Oscillations forcees de systèmes hamiltoniens, Publ. Math. Fac. Sci. Besancon, Anal. Non Lineaire Annee 1980-1981, Expose No. 4, 16 p. (1981) (French). | Zbl 0482.70020
Wu, X., Saddle point characterization and multiplicity of periodic solutions of nonautonomous second order systems, Nonlinear Anal. TMA 58 (2004), 899-907. | Zbl 1058.34053
Wu, X. P., Tang, C. L., Periodic solutions of a class of nonautonomous second order systems, J. Math. Anal. Appl. 236 (1999), 227-235. | Zbl 0971.34027
Zhao F., Wu, X., Periodic solutions for a class of non-autonomous second order systems, J. Math. Anal. Appl. 296 (2004), 422-434. | Zbl 1050.34062
Zhao F., Wu, X., Existence and multiplicity of periodic solution for non-autonomous second-order systems with linear nonlinearity, Nonlinear Anal. 60 (2005), 325-335. | Zbl 1087.34022
Xu, B., Tang, C. L., Some existence results on periodic solutions of ordinary p-Lapalcian systems, J. Math. Anal. Appl. 333 (2007), 1228-1236. | Zbl 1154.34331
Tian, Y., Ge, W., Periodic solutions of non-autonoumous second-order systems with a p-Lapalcian, Nonlinear Anal. TMA 66 (2007), 192-203.
Zhang, X., Tang, X., Periodic solutions for an ordinary p-Laplacian system, Taiwanese J. Math. (in press). | Zbl 1237.34088