Brake orbits with minimal period estimates of first-order variant subquadratic Hamiltonian systems

Xiaofei Zhang; Fanjing Wang; Xiaofei Zhang; Fanjing Wang

doi:10.3934/era.2022214

Electronic Research Archive

2022, Volume 30, Issue 11: 4220-4231. doi: 10.3934/era.2022214

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Research article

Brake orbits with minimal period estimates of first-order variant subquadratic Hamiltonian systems

Xiaofei Zhang ¹,
Fanjing Wang ^{2
,
,}

1.
School of Mathematics and Statistics, Shanxi Datong University, Datong 037009, China
2.
School of Statistics and Mathematics, Shanghai Lixin University of Accounting and Finance, Shanghai 201209, China

Received: 03 August 2022 Revised: 15 September 2022 Accepted: 15 September 2022 Published: 21 September 2022

Under a generalized subquadratic growth condition, brake orbits are guaranteed via the homological link theorem. Moreover, the minimal period estimate is given by Morse index estimate and $ L_{0} $-index estimate.
- Hamiltonian system,
- Sobolev spaces,
- weakly continuous functionals,
- $ L_{0} $-index,
- brake orbit,
- anisotropic growth
Citation: Xiaofei Zhang, Fanjing Wang. Brake orbits with minimal period estimates of first-order variant subquadratic Hamiltonian systems[J]. Electronic Research Archive, 2022, 30(11): 4220-4231. doi: 10.3934/era.2022214

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Abstract

Under a generalized subquadratic growth condition, brake orbits are guaranteed via the homological link theorem. Moreover, the minimal period estimate is given by Morse index estimate and $ L_{0} $-index estimate.

References

[1]	X. Zhang, C. Liu, Brake orbits with minimal period estimates of first-order anisotropic Hamiltonian systems, submitted for publication.
[2]	X. Zhang, F. Wang, Symmetric brake orbits with minimal period of first-order anisotropic Hamiltonian systems, submitted for publication.
[3]	S. Tang, X. Zhang, Subharmonic solutions and minimal periodic solutions of first-order variant subquadratic Hamiltonian systems, Topol. Methods Nonlinear Anal., 55 (2020), 517–532. https://doi.org/10.12775/tmna.2019.105 doi: 10.12775/tmna.2019.105
[4]	X. Zhang, C. Liu, Minimal brake orbits of first-order convex Hamiltonian systems with anisotropic growth, Partial Differ. Equations Appl., 2 (2021), 1–8. https://doi.org/10.1007/s42985-021-00104-9 doi: 10.1007/s42985-021-00104-9
[5]	A. M. Alghamdi, S. Gala, M. A. Ragusa, Regularity criterion for weak solutions to the Navier-Stokes involving one velocity and one vorticity components, Sib. Electron. Math. Rep., 19 (2022), 309–315. https://doi.org/10.33048/semi.2022.19.025 doi: 10.33048/semi.2022.19.025
[6]	D. Corona, F. Giannoni, Brake orbits for Hamiltonian systems of the classical type via geodesics in singular Finsler metrics, Adv. Nonlinear Anal., 11 (2022), 1223–1248. https://doi.org/10.1515/anona-2022-0222 doi: 10.1515/anona-2022-0222
[7]	C. Li, The study of minimal period estimates for brake orbits of autonomous subquadratic Hamiltonian systems, Acta Math. Sin. Engl. Ser., 31 (2015), 1645–1658. https://doi.org/10.1007/s10114-015-4421-3 doi: 10.1007/s10114-015-4421-3
[8]	C. Li, Brake subharmonic solutions of subquadratic Hamiltonian systems, Chin. Ann. Math. Ser. B, 37 (2016), 405–418. https://doi.org/10.1007/s11401-016-0970-8 doi: 10.1007/s11401-016-0970-8
[9]	C. Li, C. Liu, Brake subharmonic solutions of first order Hamiltonian systems, Sci. China Math., 52 (2015), 2719–2732. https://doi.org/10.1007/s11425-010-4105-5 doi: 10.1007/s11425-010-4105-5
[10]	C. Liu, Minimal period estimates for brake orbits of nonlinear symmetric Hamiltonian systems, Discrete Contin. Dyn. Syst., 27 (2010), 337–355. https://doi.org/10.48550/arXiv.0908.0029 doi: 10.48550/arXiv.0908.0029
[11]	C. Liu, D. Zhang, Iteration theory of $L$-index and multiplicity of brake orbits, J. Differ. Equations, 257 (2014), 1194–1245. https://doi.org/10.1016/j.jde.2014.05.006 doi: 10.1016/j.jde.2014.05.006
[12]	C. Liu, D. Zhang, Seifert conjecture in the even convex case, Commun. Pure Appl. Math., 67 (2014), 1563–1604. https://doi.org/10.1002/cpa.21525 doi: 10.1002/cpa.21525
[13]	Z. Liu, F. Wang, D. Zhang, Brake orbits of a reversible even Hamiltonian system near an equilibrium, Acta Math. Sin. Engl. Ser., 38 (2022), 263–280. https://doi.org/10.1007/s10114-022-0473-3 doi: 10.1007/s10114-022-0473-3
[14]	D. Zhang, Minimal period problems for brake orbits of nonlinear autonomous reversible semipositive Hamiltonian systems, Discrete Contin. Dyn. Syst., 35 (2015), 2227–2272. https://doi.org/10.48550/arXiv.1110.6915 doi: 10.48550/arXiv.1110.6915
[15]	X. Zhang, C. Liu, X. Lu, Minimal periodic problem for brake orbits of first order Hamiltonian systems, Topol. Methods Nonlinear Anal., 57 (2021), 73–87. https://doi.org/10.12775/TMNA.2020.032 doi: 10.12775/TMNA.2020.032
[16]	X. Zhang, C. Liu, Brake orbits of first order convex Hamiltonian systems with particular anisotropic growth, Acta Math. Sin. Engl. Ser., 36 (2020), 171–178. https://doi.org/10.1007/s10114-020-9043-8 doi: 10.1007/s10114-020-9043-8
[17]	Y. Long, Index Theory for Symplectic Paths with Applications, Birkhauser Verlag Basel, Boston, Berlin, 2002.
[18]	C. Liu, Index Theory in Nonlinear Analysis, Springer, Science Press, Beijing, 2019.
[19]	C. Liu, Y. Long, D. Zhang, Index iteration theory for brake orbit type solutions and applications, Anal. Theory Appl., 37 (2021), 129–156. https://doi.org/10.4208/ata.2021.pr80.05 doi: 10.4208/ata.2021.pr80.05
[20]	C. Liu, Asymptotically linear Hamiltonian systems with Lagrangian boundary conditions, Pac. J. Math., 232 (2007), 233–255. https://doi.org/10.2140/pjm.2007.232.233 doi: 10.2140/pjm.2007.232.233
[21]	C. Liu, Maslov-type index theory for symplectic paths with Lagrangian boundary conditions, Adv. Nonlinear Stud., 7 (2007), 131–161. https://doi.org/10.1515/ans-2007-0107 doi: 10.1515/ans-2007-0107
[22]	A. Abbondandolo, Morse Theory for Hamiltonian Systems, Chapman, Hall, London, 2001.
[23]	V. Benci, P. H. Rabinowitz, Critical point theorems for indefinite functionals, Invent. Math., 52 (1979), 241–273. https://doi.org/10.1007/BF01389883 doi: 10.1007/BF01389883
[24]	T. An, Z. Wang, Periodic solutions of Hamiltonian systems with anisotropic growth, Comm. Pure Appl. Anal., 9 (2010), 1069–1082. https://doi.org/10.3934/cpaa.2010.9.1069 doi: 10.3934/cpaa.2010.9.1069

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