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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 |
$ \ddot{u}(t)-L(t) u(t)+\nabla W(t,u(t)) = 0 $ |
$ L:R\rightarrow R^{N^2} $ |
$ W \in C^1(R\times R^N, R) $ |
$ t $ |
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. |
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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