| Title:
|
Periodic solutions for a class of non-autonomous Hamiltonian systems with $p(t)$-Laplacian (English) |
| Author:
|
Wang, Zhiyong |
| Author:
|
Qian, Zhengya |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
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149 |
| Issue:
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2 |
| Year:
|
2024 |
| Pages:
|
185-208 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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 $p(t)\equiv p=2$. (English) |
| Keyword:
|
auxiliary functions |
| Keyword:
|
$p(t)$-Laplacian systems |
| Keyword:
|
periodic solution |
| Keyword:
|
(C) condition |
| Keyword:
|
generalized mountain pass theorem |
| MSC:
|
34C25 |
| MSC:
|
35A15 |
| DOI:
|
10.21136/MB.2023.0096-22 |
| . |
| Date available:
|
2024-07-10T15:03:02Z |
| Last updated:
|
2024-07-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/152467 |
| . |
| Reference:
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[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. Zbl 0522.58012, MR 0713209, 10.1016/0362-546X(83)90115-3 |
| Reference:
|
[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. Zbl 0436.58006, MR 0581298 |
| Reference:
|
[3] Diening, L., Harjulehto, P., Hästö, P., Růžička, M.: Lebesgue and Sobolev Spaces with Variable Exponents.Lecture Notes in Mathematics 2017. Springer, Berlin (2011). Zbl 1222.46002, MR 2790542, 10.1007/978-3-642-18363-8 |
| Reference:
|
[4] Fan, X.-L., Fan, X.: A Knobloch-type result for $p(t)$-Laplacian systems.J. Math. Anal. Appl. 282 (2003), 453-464. Zbl 1033.34023, MR 1989103, 10.1016/S0022-247X(02)00376-1 |
| Reference:
|
[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. Zbl 1055.34082, MR 1978419, 10.1016/S0362-546X(03)00099-3 |
| Reference:
|
[6] Fei, G.: On periodic solutions of superquadratic Hamiltonian systems.Electron. J. Differ. Equ. 2002 (2002), Article ID 8, 12 pages. Zbl 0999.37039, MR 1884977 |
| Reference:
|
[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. Zbl 1118.34038, MR 2285556, 10.1016/j.jmaa.2006.05.064 |
| Reference:
|
[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. Zbl 1218.37080, MR 2764361, 10.1016/j.na.2010.10.030 |
| Reference:
|
[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. Zbl 1320.34065, MR 3294474, 10.1186/s13661-014-0260-x |
| Reference:
|
[10] Liu, C., Zhong, Y.: Infinitely many periodic solutions for ordinary $p(t)$-Laplacian differential systems.Electron Res. Arch. 30 (2022), 1653-1667. MR 4401210, 10.3934/era.2022083 |
| Reference:
|
[11] Ma, S., Zhang, Y.: Existence of infinitely many periodic solutions for ordinary $p$-Laplacian systems.J. Math. Anal. Appl. 351 (2009), 469-479. Zbl 1153.37009, MR 2472958, 10.1016/j.jmaa.2008.10.027 |
| Reference:
|
[12] Mawhin, J., Willem, M.: Critical Point Theory and Hamiltonian Systems.Applied Mathematical Sciences 74. Springer, New York (1989). Zbl 0676.58017, MR 0982267, 10.1007/978-1-4757-2061-7 |
| Reference:
|
[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. Zbl 1063.34033, MR 2073524, 10.1016/j.na.2004.03.029 |
| Reference:
|
[14] Pipan, J., Schechter, M.: Non-autonomous second order Hamiltonian systems.J. Differ. Equations 257 (2014), 351-373. Zbl 1331.37085, MR 3200374, 10.1016/j.jde.2014.03.016 |
| Reference:
|
[15] Rabinowitz, P.: On subharmonic solutions of Hamiltonian systems.Commun. Pure Appl. Math. 33 (1980), 609-633. Zbl 0425.34024, MR 0586414, 10.1002/cpa.3160330504 |
| Reference:
|
[16] Rabinowitz, P.: Minimax Methods in Critical Point Theory with Applications to Differential Equations.Regional Conference Series in Mathematics 65. AMS, Providence (1986). Zbl 0609.58002, MR 0845785, 10.1090/cbms/065 |
| Reference:
|
[17] Schechter, M.: Periodic non-autonomous second-order dynamical systems.J. Differ. Equations 223 (2006), 290-302. Zbl 1099.34042, MR 2214936, 10.1016/j.jde.2005.02.022 |
| Reference:
|
[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. Zbl 1314.34090, MR 3212230, 10.1016/j.aml.2014.04.001 |
| Reference:
|
[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. Zbl 1206.34059, MR 2651840, 10.1016/j.camwa.2010.03.039 |
| Reference:
|
[20] Tao, Z.-L., Tang, C.-L.: Periodic and subharmonic solutions of second-order Hamiltonian systems.J. Math. Anal. Appl. 293 (2004), 435-445. Zbl 1042.37047, MR 2053889, 10.1016/j.jmaa.2003.11.007 |
| Reference:
|
[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. Zbl 1116.34034, MR 2271646, 10.1016/j.na.2005.11.020 |
| Reference:
|
[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. Zbl 1171.34030, MR 2468426, 10.1016/j.na.2008.01.017 |
| Reference:
|
[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. Zbl 1401.34052, MR 3806888, 10.1016/j.crma.2018.04.014 |
| Reference:
|
[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. Zbl 1461.37067, MR 3748609, 10.1016/j.aml.2017.11.016 |
| Reference:
|
[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. Zbl 1154.34331, MR 2331727, 10.1016/j.jmaa.2006.11.051 |
| Reference:
|
[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. Zbl 1275.34060, MR 2851529, 10.1186/1687-2770-2011-33 |
| Reference:
|
[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. Zbl 1246.34042, MR 2952800, 10.36045/bbms/1331153413 |
| Reference:
|
[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. Zbl 1357.34080, MR 3575775, 10.1186/s13661-016-0720-6 |
| Reference:
|
[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. Zbl 1297.34058, MR 3072825, 10.1186/1687-2770-2013-139 |
| Reference:
|
[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. Zbl 1181.34054, MR 2505673, 10.3934/dcdsb.2009.12.251 |
| Reference:
|
[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. Zbl 0599.49031, MR 0864171, 10.1070/IM1987v029n01ABEH000958 |
| Reference:
|
[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. Zbl 0997.37039, MR 1828377, 10.1016/S0362-546X(99)00324-7 |
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