Ground state homoclinic orbits for a class of asymptotically periodic second-order Hamiltonian systems

Ying Lv; Yan-Fang Xue; Chun-Lei Tang

doi:10.3934/dcdsb.2020176

Article Contents

2021, Volume 26, Issue 3: 1627-1652. Doi: 10.3934/dcdsb.2020176

This issue Previous Article Ergodicity of stochastic damped Ostrovsky equation driven by white noise Next Article Dynamic aspects of Sprott BC chaotic system

Ground state homoclinic orbits for a class of asymptotically periodic second-order Hamiltonian systems

1.
School of Mathematics and Statistics, Southwest University, Chongqing 400715, China
2.
School of Mathematics and Statistics, Xinyang Normal University, Henan 464000, China

^* Corresponding author: Chun-Lei Tang
^* Corresponding author: Chun-Lei Tang

Received: December 31, 2018

Revised: February 29, 2020

Early access: June 2020

Published: March 2021

The first author is supported by Fundamental Research Funds for the Central Universities (XDJK2020B051) and National Natural Science Foundation of China(No. 11601438, 11971393)

Abstract Full Text(HTML) Related Papers Cited by

Abstract

In this paper, we consider a class of second-order Hamiltonian systems of the form

$ \ddot{u}(t)-L(t) u(t)+\nabla W(t,u(t)) = 0 $

where $ L:R\rightarrow R^{N^2} $ and $ W \in C^1(R\times R^N, R) $ are asymptotically periodic in $ t $ at infinity. Under the reformative perturbation conditions and weaker superquadratic conditions on the nonlinearity, the existence of a ground state homoclinic orbit is established. The main tools employed here are the local mountain pass theorem and the concentration-compactness principle.

Keywords:

Mathematics Subject Classification: Primary: 37J45, 37K05; Secondary: 58E05.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	C. O. Alves, P. C. Carrião and O. H. Miyagaki, Existence of homoclinic orbits for asymptotically periodic systems involving Duffing-like equation, Appl. Math. Lett., 16 (2003), no. 5,639–642. doi: 10.1016/S0893-9659(03)00059-4.
[2]	A. Andrzej and T. Weth, The Method of Nehari Manifold. Handbook Of Nonconvex Analysis and Applications, Int. Press, Somerville, MA, 2010,597-632.
[3]	G. Arioli and A. Szulkin, Homoclinic solution for a class of systems of second order differential equations, Topol. Methods Nonlinear Anal., 6 (1995), no. 1,189–197. doi: 10.12775/TMNA.1995.040.
[4]	P. Bartolo, V. Benci and D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with "strong" resonance at infinity, Nonlinear Anal., 7 (1983), no. 9,981–1012. doi: 10.1016/0362-546X(83)90115-3.
[5]	G. W. Chen, Superquadratic or asymptotically quadratic Hamiltonian systems: Ground state homoclinic orbits, Ann. Mat. Pura Appl. (4), 194 (2015), no. 3,903–918. doi: 10.1007/s10231-014-0403-9.
[6]	V. Coti Zelati, I. Ekeland and E. Séré, A variational approach to homoclinic orbits in Hamiltonian systems, Math. Ann., 288 (1990), no. 1,133–160. doi: 10.1007/BF01444526.
[7]	V. Coti Zelati and P. H. Rabinowitz, Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials, J. Amer. Math. Soc., 4 (1991), no. 4,693–727. doi: 10.1090/S0894-0347-1991-1119200-3.
[8]	Y. Ding, Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems, Nonlinear Anal., 25 (1995), no. 11, 1095–1113. doi: 10.1016/0362-546X(94)00229-B.
[9]	Y. Ding and C. Lee, Homoclinics for asymptotically quadratic and superquadratic Hamiltonian systems, Nonlinear Anal., 71 (2009), no. 5-6, 1395–1413. doi: 10.1016/j.na.2008.10.116.
[10]	P. L. Felmer and E. A. de B. Silva, Homoclinic and periodic orbits for Hamiltonian systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 26 (1998), no. 2,285–301.
[11]	P. Korman and A. C. Lazer, Homoclinic orbits for a class of symmetric Hamiltonian systems, Electron. J. Differential Equations, (1994), no. 1, 10 pp.
[12]	O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York-London, 1968.
[13]	P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Ⅰ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), no. 2,109–145. doi: 10.1016/S0294-1449(16)30428-0.
[14]	P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Ⅱ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), no. 4,223–283. doi: 10.1016/S0294-1449(16)30422-X.
[15]	H. F. Lins and E. A. de B. Silva, Quasilinear asymptotically periodic elliptic equations with critical growth, Nonlinear Anal., 71 (2009), no. 7-8, 2890–2905. doi: 10.1016/j.na.2009.01.171.
[16]	Z. Liu, S. Guo and Z. Zhang, Homoclinic orbits for the second-order Hamiltonian systems, Nonlinear Anal. Real World Appl., 36 (2017), 116-138. doi: 10.1016/j.nonrwa.2016.12.006.
[17]	X. Lv, S. Lu and P. Yan, Existence of homoclinic solutions for a class of second-order Hamiltonian systems, Nonlinear Anal., 72 (2010), no. 1,390–398. doi: 10.1016/j.na.2009.06.073.
[18]	Y. Lv and C.-L. Tang, Existence of even homoclinic orbits for second-order Hamiltonian systems, Nonlinear Anal., 67 (2007), no. 7, 2189–2198. doi: 10.1016/j.na.2006.08.043.
[19]	W. Omana and M. Willem, Homoclinic orbits for a class of Hamiltonian systems, Different Integral Equations, 5 (1992), no. 5, 1115–1120.
[20]	Z.-Q. Ou and C.-L. Tang, Existence of homoclinic solution for the second order Hamiltonian systems, J. Math. Anal. Appl., 291 (2004), no. 1,203–213. doi: 10.1016/j.jmaa.2003.10.026.
[21]	E. Paturel, Multiple homoclinic orbits for a class of Hamiltonian systems, Calc. Var. Partial Differential Equations, 12 (2001), no. 2,117–143. doi: 10.1007/PL00009909.
[22]	H. Poincaré, Les méthods nouvelles de la mécanique céleste, Gauthier-Villars, Paris, 1897–1899.
[23]	P. H. Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems, Proc. Roy. Soc. Edinburgh Sect., 114 (1990), no. 1-2, 33–38. doi: 10.1017/S0308210500024240.
[24]	P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, Vol. 65, American Mathematical Society, Providence, RI, 1986. doi: 10.1090/cbms/065.
[25]	P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), no. 2,270–291. doi: 10.1007/BF00946631.
[26]	P. H. Rabinowitz and K. Tanaka, Some results on connecting orbits for a class of Hamiltonian systems, Math. Z., 206 (1991), no. 3,473–499. doi: 10.1007/BF02571356.
[27]	Y. Rong and Z. Zhang, Homoclinic solutions for a class of second order Hamiltonian systems, Results Math., 61 (2012), no. 1-2,195–208. doi: 10.1007/s00025-010-0088-3.
[28]	E. Serra, M. Tarallo and S. Terracini, Subharmonic solutions to second-order differential equations with periodic nonlinearities, Nonlinear Anal., 41 (2000), 649-667. doi: 10.1016/S0362-546X(98)00302-2.
[29]	M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, Vol. 24, Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.
[30]	M. Yang and Z. Han, Infinitely many homoclinic solutions for second-order Hamiltonian systems with odd nonlinearities, Nonlinear Anal., 74 (2011), no. 7, 2635–2646. doi: 10.1016/j.na.2010.12.019.
[31]	J. Yang and F. Zhang, Infinitely many homoclinic orbits for the second-order Hamiltonian systems with super-quadratic potentials, Nonlinear Anal. Real World Appl., 10 (2009), no. 3, 1417–1423. doi: 10.1016/j.nonrwa.2008.01.013.
[32]	Q. Zhang and C. Liu, Infinitely many homoclinic solutions for second order Hamiltonian systems, Nonlinear Anal., 72 (2010), no. 2,894–903. doi: 10.1016/j.na.2009.07.021.
[33]	Z. Zhang and R. Yuan, Homoclinic solutions for a class of non-autonomous subquadratic second-order Hamiltonian systems, Nonlinear Anal., 71 (2009), no. 9, 4125–4130. doi: 10.1016/j.na.2009.02.071.