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

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March 2021, 26(3): 1627-1652. doi: 10.3934/dcdsb.2020176

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

Ying Lv ^1, , Yan-Fang Xue ^1,2, and Chun-Lei Tang ^1,2,,

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

Received January 2019 Revised March 2020 Published March 2021 Early access June 2020

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

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: Hamiltonian systems, homoclinic orbit, ground state, local mountain pass theorem, concentration-compactness principle.

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

Citation: Ying Lv, Yan-Fang Xue, Chun-Lei Tang. Ground state homoclinic orbits for a class of asymptotically periodic second-order Hamiltonian systems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1627-1652. doi: 10.3934/dcdsb.2020176

References:

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[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.

show all references

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.

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

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