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January  2011, 10(1): 269-286. doi: 10.3934/cpaa.2011.10.269

Homoclinic orbits for a class of Hamiltonian systems with superquadratic or asymptotically quadratic potentials

Jun Wang 1, , Junxiang Xu 1, and  Fubao Zhang 1,

1. 

Department of Mathematics, Southeast University, Nanjing 210096

Received  November 2009 Revised  August 2010 Published  November 2010

In this paper we study the following nonperiodic second order Hamiltonian system

$-\ddot{u}(t)+L(t)u(t)=\nabla_u R(t,u(t)), \forall (t,u)\in R\times R^N, $

where the matrix $L(t)\in C(R,R^{N^2})$ and $R(t,u)$ is asymptotically quadratic or super quadratic in $u$ as $|u|\rightarrow\infty$. Under more general assumptions on the matrix $L(t)$, if $R$ is superquadratic and even in $u$, we obtain infinitely many homoclinic orbits. On the other hand, if $R$ is asymptotically quadratic, we also prove the existence and multiplicity of homoclinic orbits for the above system.

Keywords: variational methods, Hamiltonian systems, $(C)_c$-sequence., homoclinic orbits.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Jun Wang, Junxiang Xu, Fubao Zhang. Homoclinic orbits for a class of Hamiltonian systems with superquadratic or asymptotically quadratic potentials. Communications on Pure & Applied Analysis, 2011, 10 (1) : 269-286. doi: 10.3934/cpaa.2011.10.269
References:
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Y. H. Ding, "Variational Methods for Strongly Indefinite Problems," World Scientific Press, 2008.  Google Scholar

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Y. H. Ding and B. Ruf, Solutions of a nonliear Dirac equation with external feilds, Arch. Rational Mech. Anal., 190 (2008), 57-82. doi: doi:10.1007/s00205-008-0163-z.  Google Scholar

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Y. H. Ding and M. Girardi, Infinitely many homoclinic orbits of a Hamiltonian system with symmetry, Nonlinear Anal., 38 (1999), 391-415. doi: doi:10.1016/S0362-546X(98)00204-1.  Google Scholar

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W. M. Zou and M. Schechter, "Critical Point Theory and its Applications," Springer, 2006.  Google Scholar

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V. Coti-Zelati and P. H. Rabinowitz, Homoclinic orbits for second order Hamiltonian systems possessing superqudratic potentials, J. Amer. Math. Soc., 4 (1991), 693-742. doi: doi:10.1090/S0894-0347-1991-1119200-3.  Google Scholar

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W. Omana and M. Willem, Homoclinic orbits for a class of Hamiltonian systems, Diff. and Int. Eq., 5 (1992), 1115-1120.  Google Scholar

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P. H. Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems, Proc. Roy. Soc. Edinburgh, 114 (1990), 33-38.  Google Scholar

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Z. Q. Qu and C. L. Tang, Existence of homoclinic solution for the second order Hamiltonian systems, J. Math. Anal. Appl., 291 (2004), 203-213. doi: doi:10.1016/j.jmaa.2003.10.026.  Google Scholar

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Y. Lv and C. L. Tang, Existence of even homoclinic solution for the second order Hamiltonian systems, Nonlinea Anal., 67 (2007), 2189-2198. doi: doi:10.1016/j.na.2006.08.043.  Google Scholar

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J. Yang and F. B. Zhang, Infinitely many homoclinic orbits for the second order Hamiltonian systems with superquadratic potentials, Nonlinear Anal. Real World Appl., 10 (2009), 1417-1423. doi: doi:10.1016/j.nonrwa.2008.01.013.  Google Scholar

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W. M. Zou and S. J. Li, Infinitely many homoclinic orbits for the second order Hamiltonian systems, Applied Mathematics Letter, 16 (2003), 1283-1287. doi: doi:10.1016/S0893-9659(03)90130-3.  Google Scholar

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D. E. Edmunds and W. D. Evans, "Spectral Theory and Differential Operators, Clarendon Press," Oxford, 1987.  Google Scholar

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V. Coti-Zelati, I. Ekeland and E. Séré, A variational approach to homoclinic orbits in Hamiltonian systems, Math. Ann., 228 (1990), 133-160. doi: doi:10.1007/BF01444526.  Google Scholar

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Y. H. Ding and S. J. Li, Homoclinic orbits for first order Hamiltonian systems, J. Math. Anal. Appl., 189 (1995), 585-601. doi: doi:10.1006/jmaa.1995.1037.  Google Scholar

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H. Hofer and K. Wysocki, First order elliptic systems and the existence of homoclinic orbits in Hamiltonian systems, Math. Ann., 228 (1990), 483-503. doi: doi:10.1007/BF01444543.  Google Scholar

[23]

E. Séré, Existence of infinitely many homoclinic orbits in Hamiltonian systems, Math. Z., 209 (1992), 27-42. doi: doi:10.1007/BF02570817.  Google Scholar

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K. Tanaka, Homoclinic orbits in a first order superquadratic Hamiltonian system: Conver- gence of subharmonic orbits, J. Diff. Eq., 94 (1991), 315-339. doi: doi:10.1016/0022-0396(91)90095-Q.  Google Scholar

[25]

Y. H. Ding and M. Willem, Homoclinic orbits of a Hamiltonian system, Z. Angew. Math. Phys., 50 (1999), 759-778. doi: doi:10.1007/s000330050177.  Google Scholar

[26]

A. Szulkin and W. M. Zou, Homoclinic orbits for asymptotically linear Hamiltonian systems, J. Funct. Anal., 187 (2001), 25-41. doi: doi:10.1006/jfan.2001.3798.  Google Scholar

[27]

M. Izydorek and J. Janczewska, Homoclinic solutions for a class of the second order Hamiltonian systems, J. Differential Equations, 219 (2005), 375-389. doi: doi:10.1016/j.jde.2005.06.029.  Google Scholar

[28]

M. Izydorek and J. Janczewska, Homoclinic solutions for nonautonomous second order Hamiltonian systems with a coercive potential, J. Math. Anal. Appl., 335 (2007), 1119-1127. doi: doi:10.1016/j.jmaa.2007.02.038.  Google Scholar

[29]

M. Schechter and W. M. Zou, Homoclinic orbits for Schrödinger systems, Michigan Math. J., 51 (2005), 59-71.  Google Scholar

[30]

Y. H. Ding, Multiple homoclinics in a Hamiltonian system with asymptotically or super linear terms, Communications in Contemporary Mathematics, 4 (2006), 453-480. doi: doi:10.1142/S0219199706002192.  Google Scholar

show all references

References:
[1]

Y. H. Ding, "Variational Methods for Strongly Indefinite Problems," World Scientific Press, 2008.  Google Scholar

[2]

Y. H. Ding and B. Ruf, Solutions of a nonliear Dirac equation with external feilds, Arch. Rational Mech. Anal., 190 (2008), 57-82. doi: doi:10.1007/s00205-008-0163-z.  Google Scholar

[3]

Y. H. Ding and A. Szulkin, Bound states for semilinear Schrödinger equations with sign-changing potential, Calc. Var. Partial Differential Equations, 29 (2007), 397-419. doi: doi:10.1007/s00526-006-0071-8.  Google Scholar

[4]

Y. H. Ding and L. Jeanjean, Homoclinic orbits for non periodic Hamiltonian system, J. Differential Equations, 237 (2007), 473-490. doi: doi:10.1016/j.jde.2007.03.005.  Google Scholar

[5]

Y. H. Ding, Existence and multiciplicity results for homoclinic solutions to a class of Hamiltonian systems, Nonlin. Anal. T.M.A., 25 (1995), 1095-1113. doi: doi:10.1016/0362-546X(94)00229-B.  Google Scholar

[6]

Y. H. Ding and M. Girardi, Periodic and homoclinic solutions to a class of Hamiltonian systems with the potentials changing sign, Dyn. Sys. and Appl., 2 (1993), 131-145.  Google Scholar

[7]

Y. H. Ding and M. Girardi, Infinitely many homoclinic orbits of a Hamiltonian system with symmetry, Nonlinear Anal., 38 (1999), 391-415. doi: doi:10.1016/S0362-546X(98)00204-1.  Google Scholar

[8]

M. Struwe, "Variational Methods Applications to Nolinear Partial Differential Equations and Hamiltonian Systems," Springer-Verlag, Berlin, 2000.  Google Scholar

[9]

P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations," CBMS Reg. Conf.Ser. in Math. Vol. 65 A. M. S., Providence, 1986.  Google Scholar

[10]

W. M. Zou and M. Schechter, "Critical Point Theory and its Applications," Springer, 2006.  Google Scholar

[11]

V. Coti-Zelati and P. H. Rabinowitz, Homoclinic orbits for second order Hamiltonian systems possessing superqudratic potentials, J. Amer. Math. Soc., 4 (1991), 693-742. doi: doi:10.1090/S0894-0347-1991-1119200-3.  Google Scholar

[12]

P. H. Rabinowitz and K. Tanaka, Some results on connecting orbits for a class of Hamiltonin systems, Math. Z., 206 (1990), 473-499. doi: doi:10.1007/BF02571356.  Google Scholar

[13]

W. Omana and M. Willem, Homoclinic orbits for a class of Hamiltonian systems, Diff. and Int. Eq., 5 (1992), 1115-1120.  Google Scholar

[14]

P. H. Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems, Proc. Roy. Soc. Edinburgh, 114 (1990), 33-38.  Google Scholar

[15]

Z. Q. Qu and C. L. Tang, Existence of homoclinic solution for the second order Hamiltonian systems, J. Math. Anal. Appl., 291 (2004), 203-213. doi: doi:10.1016/j.jmaa.2003.10.026.  Google Scholar

[16]

Y. Lv and C. L. Tang, Existence of even homoclinic solution for the second order Hamiltonian systems, Nonlinea Anal., 67 (2007), 2189-2198. doi: doi:10.1016/j.na.2006.08.043.  Google Scholar

[17]

J. Yang and F. B. Zhang, Infinitely many homoclinic orbits for the second order Hamiltonian systems with superquadratic potentials, Nonlinear Anal. Real World Appl., 10 (2009), 1417-1423. doi: doi:10.1016/j.nonrwa.2008.01.013.  Google Scholar

[18]

W. M. Zou and S. J. Li, Infinitely many homoclinic orbits for the second order Hamiltonian systems, Applied Mathematics Letter, 16 (2003), 1283-1287. doi: doi:10.1016/S0893-9659(03)90130-3.  Google Scholar

[19]

D. E. Edmunds and W. D. Evans, "Spectral Theory and Differential Operators, Clarendon Press," Oxford, 1987.  Google Scholar

[20]

V. Coti-Zelati, I. Ekeland and E. Séré, A variational approach to homoclinic orbits in Hamiltonian systems, Math. Ann., 228 (1990), 133-160. doi: doi:10.1007/BF01444526.  Google Scholar

[21]

Y. H. Ding and S. J. Li, Homoclinic orbits for first order Hamiltonian systems, J. Math. Anal. Appl., 189 (1995), 585-601. doi: doi:10.1006/jmaa.1995.1037.  Google Scholar

[22]

H. Hofer and K. Wysocki, First order elliptic systems and the existence of homoclinic orbits in Hamiltonian systems, Math. Ann., 228 (1990), 483-503. doi: doi:10.1007/BF01444543.  Google Scholar

[23]

E. Séré, Existence of infinitely many homoclinic orbits in Hamiltonian systems, Math. Z., 209 (1992), 27-42. doi: doi:10.1007/BF02570817.  Google Scholar

[24]

K. Tanaka, Homoclinic orbits in a first order superquadratic Hamiltonian system: Conver- gence of subharmonic orbits, J. Diff. Eq., 94 (1991), 315-339. doi: doi:10.1016/0022-0396(91)90095-Q.  Google Scholar

[25]

Y. H. Ding and M. Willem, Homoclinic orbits of a Hamiltonian system, Z. Angew. Math. Phys., 50 (1999), 759-778. doi: doi:10.1007/s000330050177.  Google Scholar

[26]

A. Szulkin and W. M. Zou, Homoclinic orbits for asymptotically linear Hamiltonian systems, J. Funct. Anal., 187 (2001), 25-41. doi: doi:10.1006/jfan.2001.3798.  Google Scholar

[27]

M. Izydorek and J. Janczewska, Homoclinic solutions for a class of the second order Hamiltonian systems, J. Differential Equations, 219 (2005), 375-389. doi: doi:10.1016/j.jde.2005.06.029.  Google Scholar

[28]

M. Izydorek and J. Janczewska, Homoclinic solutions for nonautonomous second order Hamiltonian systems with a coercive potential, J. Math. Anal. Appl., 335 (2007), 1119-1127. doi: doi:10.1016/j.jmaa.2007.02.038.  Google Scholar

[29]

M. Schechter and W. M. Zou, Homoclinic orbits for Schrödinger systems, Michigan Math. J., 51 (2005), 59-71.  Google Scholar

[30]

Y. H. Ding, Multiple homoclinics in a Hamiltonian system with asymptotically or super linear terms, Communications in Contemporary Mathematics, 4 (2006), 453-480. doi: doi:10.1142/S0219199706002192.  Google Scholar

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