Existence and concentration of homoclinic orbits for first order Hamiltonian systems

Tianfang Wang; Wen Zhang; Tianfang Wang; Wen Zhang

doi:10.3934/cam.2024006

Communications in Analysis and Mechanics

2024, Volume 16, Issue 1: 121-146. doi: 10.3934/cam.2024006

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Research article

Existence and concentration of homoclinic orbits for first order Hamiltonian systems

Tianfang Wang ^{1,2
,},
Wen Zhang ^{3
,
,}

1.
School of Educational Sciences, Baotou Teachers' College, 014030 Baotou, Inner Mongolia, China
2.
Faculty of Applied Mathematics, AGH University of Science and Technology, 30-059 Kraków, Poland
3.
College of Science, Hunan University of Technology and Business, 410205 Changsha, Hunan, China

Received: 12 September 2023 Revised: 18 December 2023 Accepted: 12 January 2024 Published: 23 January 2024
37J45, 70H05, 58E50

This paper is concerned with the following first-order Hamiltonian system

$ \begin{equation} \nonumber \dot{z} = \mathscr{J}H_{z}(t, z), \end{equation} $

where the Hamiltonian function $ H(t, z) = \frac{1}{2}Lz\cdot z+A(\epsilon t)G(|z|) $ and $ \epsilon > 0 $ is a small parameter. Under some natural conditions, we obtain a new existence result for ground state homoclinic orbits by applying variational methods. Moreover, the concentration behavior and exponential decay of these ground state homoclinic orbits are also investigated.
- Hamiltonian system,
- ground state homoclinic orbits,
- concentration
Citation: Tianfang Wang, Wen Zhang. Existence and concentration of homoclinic orbits for first order Hamiltonian systems[J]. Communications in Analysis and Mechanics, 2024, 16(1): 121-146. doi: 10.3934/cam.2024006

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Abstract

This paper is concerned with the following first-order Hamiltonian system

$ \begin{equation} \nonumber \dot{z} = \mathscr{J}H_{z}(t, z), \end{equation} $

where the Hamiltonian function $ H(t, z) = \frac{1}{2}Lz\cdot z+A(\epsilon t)G(|z|) $ and $ \epsilon > 0 $ is a small parameter. Under some natural conditions, we obtain a new existence result for ground state homoclinic orbits by applying variational methods. Moreover, the concentration behavior and exponential decay of these ground state homoclinic orbits are also investigated.

References

[1]	T. Bartsch, A. Szulkin, Hamiltonian systems: Periodic and homoclinic solutions by variational methods, in: Handbook of Differential Equations: Ordinary Differential Equations, Vol. Ⅱ, Elsevier B. V., Amsterdam, 2005, 77–146. https://doi.org/10.1007/s00605-004-0289-5
[2]	J. Mawhin, M. Willem, Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences, 74, Springer-Verlag, New York, 1989.
[3]	A. Brugnoli, G. Haine, D. Matignon, Stokes-Dirac structures for distributed parameter port-Hamiltonian systems: An analytical viewpoint, Commun. Anal. Mech., 15 (2023), 362–387. https://doi.org/10.3934/cam.2023018 doi: 10.3934/cam.2023018
[4]	P. H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math., 31 (1978), 157–184. https://doi.org/10.1002/cpa.3160310203 doi: 10.1002/cpa.3160310203
[5]	V. Coti-Zelati, I. Ekeland, E. Séré, A variational approach to homoclinic orbits in Hamiltonian systems, Math. Ann. 228 (1990), 133–160. https://doi.org/10.1007/BF01444526 doi: 10.1007/BF01444526
[6]	V. Coti Zelati, P. H. Rabinowitz, Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials, J. Amer. Math. Soc., 4 (1991), 693–727. https://doi.org/10.2307/2939286 doi: 10.2307/2939286
[7]	E. Séré, Existence of infinitely many homoclinic orbits in Hamiltonian systems, Math. Z., 209 (1992), 27–42. https://doi.org/10.1007/BF02570817 doi: 10.1007/BF02570817
[8]	H. Hofer, K. Wysocki, First order ellipic systems and the existence of homoclinic orbits in Hamiltonian systems, Math. Ann., 228 (1990), 483–503. https://doi.org/10.1007/BF01444543 doi: 10.1007/BF01444543
[9]	K. Tanaka, Homoclinic orbits in a first order superquadratic Hamiltonian system: Convergence of subharmonic orbits, J. Differential Equations, 94 (1991), 315–339. https://doi.org/10.1016/0022-0396(91)90095-Q doi: 10.1016/0022-0396(91)90095-Q
[10]	S. A. Rashkovskiy, Quantization of Hamiltonian and non-Hamiltonian systems, Commun. Anal. Mech., 15 (2023), 267–288. https://doi.org/10.3934/cam.2023014 doi: 10.3934/cam.2023014
[11]	W. Kryszewski, A. Szulkin, Generalized linking theorem with an application to semilinear Schrödinger equation, Adv. Differential Equations, 3 (1998), 441–472. https://doi.org/10.57262/ade/1366399849 doi: 10.57262/ade/1366399849
[12]	T. Bartsch, Y. Ding, Deformation theorems on non-metrizable vector spaces and applications to critical point theory, Math. Nach., 279 (2006), 1267–1288. https://doi.org/10.1002/mana.200410420 doi: 10.1002/mana.200410420
[13]	G. Arioli, A. Szulkin, Homoclinic solutions of Hamiltonian systems with symmetry, J. Differential Equations, 158 (1999), 291–313. https://doi.org/10.1006/jdeq.1999.3639 doi: 10.1006/jdeq.1999.3639
[14]	G. Chen, S. Ma, Homoclinic orbits of superlinear Hamiltonian system, Proc. Amer. Math. Soc., 139 (2011), 3973–3983. https://doi.org/10.1090/S0002-9939-2011-11185-7 doi: 10.1090/S0002-9939-2011-11185-7
[15]	Y. Ding, M. Willem, Homoclinic orbits of a Hamiltonian system, Z. Angew. Math. Phys., 50 (1999), 759–778. https://doi.org/10.1007/s000330050177 doi: 10.1007/s000330050177
[16]	Y. Ding, Multiple homoclinics in a Hamiltonian system with asymptotically or super linear terms, Commun. Contemp. Math., 4 (2006), 453–480. https://doi.org/10.1142/S0219199706002192 doi: 10.1142/S0219199706002192
[17]	Y. Ding, M. Girardi, Infinitely many homoclinic orbits of a Hamiltonian system with symmetry, Nonlinear Anal., 38 (1999), 391–415. https://doi.org/10.1016/S0362-546X(98)00204-1 doi: 10.1016/S0362-546X(98)00204-1
[18]	W. Zhang, G. Yang, F. Liao, Homoclinic orbits for first-order Hamiltonian system with local super-quadratic growth condition, Complex Var. Elliptic Equ., 67 (2022), 988–1011. https://doi.org/10.1080/17476933.2020.1857373 doi: 10.1080/17476933.2020.1857373
[19]	A. Szulkin, W. Zou, Homoclinic orbits for asymptotically linear Hamiltonian systems, J. Funct. Anal., 187 (2001), 25–41. https://doi.org/10.1006/jfan.2001.3798 doi: 10.1006/jfan.2001.3798
[20]	J. Sun, J. Chu, Z. Feng, Homoclinic orbits for first order periodic Hamiltonian systems with spectrum point zero, Discrete Contin. Dyn. Syst., 33 (2013), 3807–3824. https://doi.org/10.3934/dcds.2013.33.3807 doi: 10.3934/dcds.2013.33.3807
[21]	Y. Ding, S. Li, Homoclinic orbits for first order Hamiltonian systems, J. Math. Anal. Appl., 189 (1995), 585–601. https://doi.org/10.1006/jmaa.1995.1037 doi: 10.1006/jmaa.1995.1037
[22]	Q. Zhang, C. Liu, Homoclinic orbits for a class of first order nonperiodic Hamiltonian systems, Nonlinear Anal.: RWA, 41 (2018), 34–52. https://doi.org/10.1016/j.nonrwa.2017.10.002 doi: 10.1016/j.nonrwa.2017.10.002
[23]	Y. Ding, L. Jeanjean, Homoclinic orbits for nonperiodic Hamiltonian system, J. Differential Equations, 237 (2007), 473–490. https://doi.org/10.1016/j.jde.2007.03.005 doi: 10.1016/j.jde.2007.03.005
[24]	Y. Ding, C. Lee, Existence and exponential decay of homoclinics in a nonperiodic superquadrtic Hamiltonian system, J. Differential Equations, 246 (2009), 2829–2848. https://doi.org/10.1016/j.jde.2008.12.013 doi: 10.1016/j.jde.2008.12.013
[25]	W. Zhang, J. Zhang, X. Tang, Ground state Homoclinic orbits for first-order Hamiltonian system, Bull. Malays. Math. Sci. Soc., 43 (2020), 1163–1182. https://doi.org/10.1007/s40840-019-00734-8 doi: 10.1007/s40840-019-00734-8
[26]	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
[27]	Q. Li, J. Nie, W. Zhang, Existence and asymptotics of normalized ground states for a Sobolev critical Kirchhoff equation, J. Geom. Anal., 33 (2023), 126. https://doi.org/10.1007/s12220-022-01171-z doi: 10.1007/s12220-022-01171-z
[28]	Q. Li, V. D. Radulescu, W. Zhang, Normalized ground states for the Sobolev critical Schrödinger equation with at least mass critical growth, Nonlinearity, 37 (2024), 025018. https://doi.org/10.1088/1361-6544/ad1b8b doi: 10.1088/1361-6544/ad1b8b
[29]	N. S. Papageorgiou, V. D. Rădulescu, W. Zhang, Global existence and multiplicity for nonlinear Robin eigenvalue problems, Results Math., 78 (2023), 133. https://doi.org/10.1007/s00025-023-01912-8 doi: 10.1007/s00025-023-01912-8
[30]	N. S. Papageorgiou, J. Zhang, W. Zhang, Solutions with sign information for noncoercive double phase equations, J. Geom. Anal., 34 (2024), 14. https://doi.org/10.1007/s12220-023-01463-y doi: 10.1007/s12220-023-01463-y
[31]	D. Qin, X. Tang, J. Zhang, Ground states for planar Hamiltonian elliptic systems with critical exponential growth, J. Differential Equations, 308 (2022), 130–159. https://doi.org/10.1016/j.jde.2021.10.063 doi: 10.1016/j.jde.2021.10.063
[32]	J. Zhang, W. Zhang, Semiclassical states for coupled nonlinear Schrödinger system with competing potentials, J. Geom. Anal., 32 (2022), 114. https://doi.org/10.1007/s12220-022-00870-x doi: 10.1007/s12220-022-00870-x
[33]	C. O. Alves, G. F. Germano, Existence and concentration of ground state solution for a class of indefinite variational problem, Commun. Pure Appl. Anal., 19 (2020), 2887–2906. https://doi.org/10.3934/cpaa.2020126 doi: 10.3934/cpaa.2020126
[34]	A. Szulkin, T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257 (2009), 3802–3822. https://doi.org/10.1016/j.jfa.2009.09.013 doi: 10.1016/j.jfa.2009.09.013
[35]	J. Zhang, W. Zhang, X. Tang, Ground state solutions for Hamiltonian elliptic system with inverse square potential, Discrete Contin. Dyn. Syst., 37 (2017), 4565–4583. https://doi.org/10.3934/dcds.2017195 doi: 10.3934/dcds.2017195
[36]	J. Zhang, W. Zhang, F. Zhao, Existence and exponential decay of ground-state solutions for a nonlinear Dirac equation, Z. Angew. Math. Phys., 69 (2018), 116. https://doi.org/10.1007/s00033-018-1009-7 doi: 10.1007/s00033-018-1009-7
[37]	M. J. Esteban, E. Séré, Stationary states of nonlinear Dirac equations: a variational approach, Commun. Math. Phys., 171 (1995), 323–350. https://doi.org/10.1007/BF02099273 doi: 10.1007/BF02099273
[38]	B. Simon, Schrödinger semigroups, Bull. Am. Math. Soc., 7 (1982), 447–526. https://doi.org/10.1090/S0273-0979-1982-15041-8 doi: 10.1090/S0273-0979-1982-15041-8

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