Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
auxiliary functions; $p(t)$-Laplacian systems; periodic solution; (C) condition; generalized mountain pass theorem
auxiliary functions; $p(t)$-Laplacian systems; periodic solution; (C) condition; generalized mountain pass theorem
Summary:
We investigate the existence of infinitely many periodic solutions for the $p(t)$-Laplacian Hamiltonian systems. By virtue of several auxiliary functions, we obtain a series of new super-$p^+$ growth and asymptotic-$p^+$ growth conditions. Using the minimax methods in critical point theory, some multiplicity theorems are established, which unify and generalize some known results in the literature. Meanwhile, we also present an example to illustrate our main results are new even in the case $p(t)\equiv p=2$.
References:
[1] Bartolo, P., Benci, V., Fortunato, D.: Abstract critical point theorems and applications to some problems with "strong" resonance at infinity. Nonlinear Anal., Theory Methods Appl. 7 (1983), 981-1012. DOI 10.1016/0362-546X(83)90115-3 | MR 0713209 | Zbl 0522.58012
[2] Cerami, G.: An existence criterion for the critical points on unbounded manifolds. Ist. Lombardo Accad. Sci. Lett., Rend., Sez. A 112 (1978), 332-336 Italian. MR 0581298 | Zbl 0436.58006
[3] Diening, L., Harjulehto, P., Hästö, P., Růžička, M.: Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics 2017. Springer, Berlin (2011). DOI 10.1007/978-3-642-18363-8 | MR 2790542 | Zbl 1222.46002
[4] Fan, X.-L., Fan, X.: A Knobloch-type result for $p(t)$-Laplacian systems. J. Math. Anal. Appl. 282 (2003), 453-464. DOI 10.1016/S0022-247X(02)00376-1 | MR 1989103 | Zbl 1033.34023
[5] Faraci, F., Livrea, R.: Infinitely many periodic solutions for a second-order nonautonomous system. Nonlinear Anal., Theory Methods Appl., Ser. A 54 (2003), 417-429. DOI 10.1016/S0362-546X(03)00099-3 | MR 1978419 | Zbl 1055.34082
[6] Fei, G.: On periodic solutions of superquadratic Hamiltonian systems. Electron. J. Differ. Equ. 2002 (2002), Article ID 8, 12 pages. MR 1884977 | Zbl 0999.37039
[7] Jiang, Q., Tang, C.-L.: Periodic and subharmonic solutions of a class of subquadratic second-order Hamiltonian systems. J. Math. Anal. Appl. 328 (2007), 380-389. DOI 10.1016/j.jmaa.2006.05.064 | MR 2285556 | Zbl 1118.34038
[8] Li, C., Ou, Z.-Q., Tang, C.-L.: Three periodic solutions for $p$-Hamiltonian systems. Nonlinear Anal., Theory Methods Appl., Ser. A 74 (2011), 1596-1606. DOI 10.1016/j.na.2010.10.030 | MR 2764361 | Zbl 1218.37080
[9] Lian, H., Wang, D., Bai, Z., Agarwal, R. P.: Periodic and subharmonic solutions for a class of second-order $p$-Laplacian Hamiltonian systems. Bound. Value Probl. 2014 (2014), Article ID 260, 15 pages. DOI 10.1186/s13661-014-0260-x | MR 3294474 | Zbl 1320.34065
[10] Liu, C., Zhong, Y.: Infinitely many periodic solutions for ordinary $p(t)$-Laplacian differential systems. Electron Res. Arch. 30 (2022), 1653-1667. DOI 10.3934/era.2022083 | MR 4401210
[11] Ma, S., Zhang, Y.: Existence of infinitely many periodic solutions for ordinary $p$-Laplacian systems. J. Math. Anal. Appl. 351 (2009), 469-479. DOI 10.1016/j.jmaa.2008.10.027 | MR 2472958 | Zbl 1153.37009
[12] Mawhin, J., Willem, M.: Critical Point Theory and Hamiltonian Systems. Applied Mathematical Sciences 74. Springer, New York (1989). DOI 10.1007/978-1-4757-2061-7 | MR 0982267 | Zbl 0676.58017
[13] Ou, Z.-Q., Tang, C.-L.: Periodic and subharmonic solutions for a class of superquadratic Hamiltonian systems. Nonlinear Anal., Theory Methods Appl., Ser. A 58 (2004), 245-258. DOI 10.1016/j.na.2004.03.029 | MR 2073524 | Zbl 1063.34033
[14] Pipan, J., Schechter, M.: Non-autonomous second order Hamiltonian systems. J. Differ. Equations 257 (2014), 351-373. DOI 10.1016/j.jde.2014.03.016 | MR 3200374 | Zbl 1331.37085
[15] Rabinowitz, P.: On subharmonic solutions of Hamiltonian systems. Commun. Pure Appl. Math. 33 (1980), 609-633. DOI 10.1002/cpa.3160330504 | MR 0586414 | Zbl 0425.34024
[16] Rabinowitz, P.: Minimax Methods in Critical Point Theory with Applications to Differential Equations. Regional Conference Series in Mathematics 65. AMS, Providence (1986). DOI 10.1090/cbms/065 | MR 0845785 | Zbl 0609.58002
[17] Schechter, M.: Periodic non-autonomous second-order dynamical systems. J. Differ. Equations 223 (2006), 290-302. DOI 10.1016/j.jde.2005.02.022 | MR 2214936 | Zbl 1099.34042
[18] Tang, C.-L., Wu, X.-P.: Periodic solutions for a class of new superquadratic second order Hamiltonian systems. Appl. Math. Lett. 34 (2014), 65-71. DOI 10.1016/j.aml.2014.04.001 | MR 3212230 | Zbl 1314.34090
[19] Tang, X. H., Jiang, J.: Existence and multiplicity of periodic solutions for a class of second-order Hamiltonian systems. Comput. Math. Appl. 59 (2010), 3646-3655. DOI 10.1016/j.camwa.2010.03.039 | MR 2651840 | Zbl 1206.34059
[20] Tao, Z.-L., Tang, C.-L.: Periodic and subharmonic solutions of second-order Hamiltonian systems. J. Math. Anal. Appl. 293 (2004), 435-445. DOI 10.1016/j.jmaa.2003.11.007 | MR 2053889 | Zbl 1042.37047
[21] Tian, Y., Ge, W.: Periodic solutions of non-autonomous second-order systems with a $p$-Laplacian. Nonlinear Anal., Theory Methods Appl., Ser. A 66 (2007), 192-203. DOI 10.1016/j.na.2005.11.020 | MR 2271646 | Zbl 1116.34034
[22] Wang, X.-J., Yuan, R.: Existence of periodic solutions for $p(t)$-Laplacian systems. Nonlinear Anal., Theory Methods Appl., Ser. A 70 (2009), 866-880. DOI 10.1016/j.na.2008.01.017 | MR 2468426 | Zbl 1171.34030
[23] Wang, Z., Zhang, J.: Existence of periodic solutions for a class of damped vibration problems. C. R., Math., Acad. Sci. Paris 356 (2018), 597-612. DOI 10.1016/j.crma.2018.04.014 | MR 3806888 | Zbl 1401.34052
[24] Wang, Z., Zhang, J.: New existence results on periodic solutions of non-autonomous second order Hamiltonian systems. Appl. Math. Lett. 79 (2018), 43-50. DOI 10.1016/j.aml.2017.11.016 | MR 3748609 | Zbl 1461.37067
[25] Xu, B., Tang, C.-L.: Some existence results on periodic solutions of ordinary $p$-Laplacian systems. J. Math. Anal. Appl. 333 (2007), 1228-1236. DOI 10.1016/j.jmaa.2006.11.051 | MR 2331727 | Zbl 1154.34331
[26] Zhang, L., Tang, X. H., Chen, J.: Infinitely many periodic solutions for some second-order differential systems with $p(t)$-Laplacian. Bound. Value Probl. 2011 (2011), Article ID 33, 15 pages. DOI 10.1186/1687-2770-2011-33 | MR 2851529 | Zbl 1275.34060
[27] Zhang, Q., Tang, X. H.: On the existence of infinitely many periodic solutions for second-order ordinary $p$-Laplacian systems. Bull. Belg. Math. Soc. - Simon Stevin 19 (2012), 121-136. DOI 10.36045/bbms/1331153413 | MR 2952800 | Zbl 1246.34042
[28] Zhang, S.: Periodic solutions for a class of second order Hamiltonian systems with $p(t)$-Laplacian. Bound. Value Probl. 2016 (2016), Article ID 211, 20 pages. DOI 10.1186/s13661-016-0720-6 | MR 3575775 | Zbl 1357.34080
[29] Zhang, X., Tang, X.: Existence of subharmonic solutions for non-quadratic second-order Hamiltonian systems. Bound. Value Probl. 2013 (2013), Article ID 139, 25 pages. DOI 10.1186/1687-2770-2013-139 | MR 3072825 | Zbl 1297.34058
[30] Zhang, Y., Ma, S.: Some existence results on periodic and subharmonic solutions of ordinary $p$-Laplacian systems. Discrete Contin. Dyn. Syst., Ser. B 12 (2009), 251-260. DOI 10.3934/dcdsb.2009.12.251 | MR 2505673 | Zbl 1181.34054
[31] Zhikov, V. V.: Averaging of functionals of the calculus of variations and elasticity theory. Math. USSR, Izv. 29 (1987), 33-66 translation from Izv. Akad. Nauk SSSR, Ser. Mat. 50 1986 675-710. DOI 10.1070/IM1987v029n01ABEH000958 | MR 0864171 | Zbl 0599.49031
[32] Zou, W.: Multiple solutions for second-order Hamiltonian systems via computation of the critical groups. Nonlinear Anal., Theory Methods Appl., Ser. A 44 (2001), 975-989. DOI 10.1016/S0362-546X(99)00324-7 | MR 1828377 | Zbl 0997.37039
Search
Partner of
