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Homoclinic orbits of first-order superquadratic Hamiltonian systems
1. | Département de Mathématiques, Université de Sherbrooke, Sherbrooke, Québec, J1K 2R1, Canada |
References:
[1] |
R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. |
[2] |
T. Bartsch and A. Szulkin, Hamiltonian Systems: Periodic and Homoclinic Solutions by Variational Methods, Elsevier B. V., Amsterdam, 2005. |
[3] |
C. J. Batkam and F. Colin, Generalized fountain theorem and applications to strongly indefinite semilinear problems, J. Math. Anal. Appl., 405 (2013), 438-452.
doi: 10.1016/j.jmaa.2013.04.018. |
[4] |
C. J. Batkam and F. Colin, On multiple solutions of a semilinear Schrödinger equation with periodic potential, Nonlinear Anal., 84 (2013), 39-49.
doi: 10.1016/j.na.2013.02.006. |
[5] |
H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.
doi: 10.2307/2044999. |
[6] |
G. Chen and S. Ma, Homoclinic orbits of superlinear Hamiltonian systems, Proc. Amer. Math. Soc., 139 (2011), 3973-3983.
doi: 10.1090/S0002-9939-2011-11185-7. |
[7] |
V. Coti-Zelati, I. Ekeland and E. Séré, A variational approach to homoclinic orbits in Hamiltonian systems, Math. Ann., 228 (1990), 133-160.
doi: 10.1007/BF01444526. |
[8] |
Y. H. Ding, Multiple homoclinics in a Hamiltonian system with asymptotically or super linear terms, Commun. Contemp. Math., 8 (2006), 453-480.
doi: 10.1142/S0219199706002192. |
[9] |
Y. H. Ding and M. Girardi, Infinitely many homoclinic orbits of a Hamiltonian system with symmetry, Nonlinear Anal., 38 (1999), 391-415.
doi: 10.1016/S0362-546X(98)00204-1. |
[10] |
L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer type problem set on $\mathbbR^N$, Proc. Roy. Soc. Edinburgh A, 129 (1999), 787-809.
doi: 10.1017/S0308210500013147. |
[11] |
W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to a semilinear Schrödinger equation, Adv. Differ. Equat., 3 (1998), 441-472. |
[12] |
P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II, Ann. Inst. H. Poincaré, 1 (1984), 223-283. |
[13] |
A. Mao and S. Luan, Critical points theorems concerning strongly indefinite functionals and infinite many periodic solutions for a class of Hamiltonian systems, Appl. Math. Comput., 214 (2009), 187-200.
doi: 10.1016/j.amc.2009.03.084. |
[14] |
A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math., 73 (2005), 259-287.
doi: 10.1007/s00032-005-0047-8. |
[15] |
P. H. Rabinowitz and K. Tanaka, Some results on connecting orbits for a class of Hamiltonian systems, Math. Z., 206 (1991), 473-499.
doi: 10.1007/BF02571356. |
[16] |
E. Séré, Existence of infinitely many homoclinic orbits in Hamiltonian systems, Math. Z., 209 (1992), 27-42.
doi: 10.1007/BF02570817. |
[17] |
M. Schechter and W. Zou, Weak linking theorems and Schrödinger equations with critical Sobolev exponent, ESAIM Control Optim. Calc. Var., 9 (2003), 601-619.
doi: 10.1051/cocv:2003029. |
[18] |
M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Second Edition, Springer-Verlag, Berlin, 1996. |
[19] |
C. A. Stuart, Bifurcation into spectral gaps, Bull. Belg. Math. Soc., Supplement, (1995). |
[20] |
A. Szulkin and T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257 (2009), 3802-3822.
doi: 10.1016/j.jfa.2009.09.013. |
[21] |
A. Szulkin and T. Weth, The Method of Nehari Manifold, Int. Press, Somerville, MA, 2010. |
[22] |
A. Szulkin and W. Zou, Homoclinic orbits for asymptotically linear Hamiltonian systems, J. Funct. Anal., 187 (2001), 25-41.
doi: 10.1006/jfan.2001.3798. |
[23] |
K. Tanaka, Homoclinic orbits in a first order super-quadratic Hamiltonian system: Convergence of subharmonic orbits, J. Differential Equations, 94 (1991), 315-339.
doi: 10.1016/0022-0396(91)90095-Q. |
[24] |
J. Wang, J. X. Xu and F. B. Zhang, Homoclinic orbits for superlinear Hamiltonian systems without Ambrosetti-Rabinowitz growth condition, Discrete Contin. Dyn. Syst. Ser. A, 27 (2010), 1241-1257.
doi: 10.3934/dcds.2010.27.1241. |
[25] |
M. Willem, Minimax Theorems, Birkhauser, Boston, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[26] |
M. Yang, Ground state solutions for a periodic Schrödinger equation with superlinear nonlinearities, Nonlinear Anal., 72 (2010), 2620-2627.
doi: 10.1016/j.na.2009.11.009. |
[27] |
W. Zou, Variant fountain theorems and their applications, Manuscripta Math., 104 (2001), 343-358.
doi: 10.1007/s002290170032. |
show all references
References:
[1] |
R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. |
[2] |
T. Bartsch and A. Szulkin, Hamiltonian Systems: Periodic and Homoclinic Solutions by Variational Methods, Elsevier B. V., Amsterdam, 2005. |
[3] |
C. J. Batkam and F. Colin, Generalized fountain theorem and applications to strongly indefinite semilinear problems, J. Math. Anal. Appl., 405 (2013), 438-452.
doi: 10.1016/j.jmaa.2013.04.018. |
[4] |
C. J. Batkam and F. Colin, On multiple solutions of a semilinear Schrödinger equation with periodic potential, Nonlinear Anal., 84 (2013), 39-49.
doi: 10.1016/j.na.2013.02.006. |
[5] |
H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.
doi: 10.2307/2044999. |
[6] |
G. Chen and S. Ma, Homoclinic orbits of superlinear Hamiltonian systems, Proc. Amer. Math. Soc., 139 (2011), 3973-3983.
doi: 10.1090/S0002-9939-2011-11185-7. |
[7] |
V. Coti-Zelati, I. Ekeland and E. Séré, A variational approach to homoclinic orbits in Hamiltonian systems, Math. Ann., 228 (1990), 133-160.
doi: 10.1007/BF01444526. |
[8] |
Y. H. Ding, Multiple homoclinics in a Hamiltonian system with asymptotically or super linear terms, Commun. Contemp. Math., 8 (2006), 453-480.
doi: 10.1142/S0219199706002192. |
[9] |
Y. H. Ding and M. Girardi, Infinitely many homoclinic orbits of a Hamiltonian system with symmetry, Nonlinear Anal., 38 (1999), 391-415.
doi: 10.1016/S0362-546X(98)00204-1. |
[10] |
L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer type problem set on $\mathbbR^N$, Proc. Roy. Soc. Edinburgh A, 129 (1999), 787-809.
doi: 10.1017/S0308210500013147. |
[11] |
W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to a semilinear Schrödinger equation, Adv. Differ. Equat., 3 (1998), 441-472. |
[12] |
P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II, Ann. Inst. H. Poincaré, 1 (1984), 223-283. |
[13] |
A. Mao and S. Luan, Critical points theorems concerning strongly indefinite functionals and infinite many periodic solutions for a class of Hamiltonian systems, Appl. Math. Comput., 214 (2009), 187-200.
doi: 10.1016/j.amc.2009.03.084. |
[14] |
A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math., 73 (2005), 259-287.
doi: 10.1007/s00032-005-0047-8. |
[15] |
P. H. Rabinowitz and K. Tanaka, Some results on connecting orbits for a class of Hamiltonian systems, Math. Z., 206 (1991), 473-499.
doi: 10.1007/BF02571356. |
[16] |
E. Séré, Existence of infinitely many homoclinic orbits in Hamiltonian systems, Math. Z., 209 (1992), 27-42.
doi: 10.1007/BF02570817. |
[17] |
M. Schechter and W. Zou, Weak linking theorems and Schrödinger equations with critical Sobolev exponent, ESAIM Control Optim. Calc. Var., 9 (2003), 601-619.
doi: 10.1051/cocv:2003029. |
[18] |
M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Second Edition, Springer-Verlag, Berlin, 1996. |
[19] |
C. A. Stuart, Bifurcation into spectral gaps, Bull. Belg. Math. Soc., Supplement, (1995). |
[20] |
A. Szulkin and T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257 (2009), 3802-3822.
doi: 10.1016/j.jfa.2009.09.013. |
[21] |
A. Szulkin and T. Weth, The Method of Nehari Manifold, Int. Press, Somerville, MA, 2010. |
[22] |
A. Szulkin and W. Zou, Homoclinic orbits for asymptotically linear Hamiltonian systems, J. Funct. Anal., 187 (2001), 25-41.
doi: 10.1006/jfan.2001.3798. |
[23] |
K. Tanaka, Homoclinic orbits in a first order super-quadratic Hamiltonian system: Convergence of subharmonic orbits, J. Differential Equations, 94 (1991), 315-339.
doi: 10.1016/0022-0396(91)90095-Q. |
[24] |
J. Wang, J. X. Xu and F. B. Zhang, Homoclinic orbits for superlinear Hamiltonian systems without Ambrosetti-Rabinowitz growth condition, Discrete Contin. Dyn. Syst. Ser. A, 27 (2010), 1241-1257.
doi: 10.3934/dcds.2010.27.1241. |
[25] |
M. Willem, Minimax Theorems, Birkhauser, Boston, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[26] |
M. Yang, Ground state solutions for a periodic Schrödinger equation with superlinear nonlinearities, Nonlinear Anal., 72 (2010), 2620-2627.
doi: 10.1016/j.na.2009.11.009. |
[27] |
W. Zou, Variant fountain theorems and their applications, Manuscripta Math., 104 (2001), 343-358.
doi: 10.1007/s002290170032. |
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