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

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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

Abstract
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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 and Applied Analysis, 2011, 10 (1) : 269-286. doi: 10.3934/cpaa.2011.10.269

References:

[1]	Y. H. Ding, "Variational Methods for Strongly Indefinite Problems," World Scientific Press, 2008.
[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.
[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.
[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.
[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.
[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.
[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.
[8]	M. Struwe, "Variational Methods Applications to Nolinear Partial Differential Equations and Hamiltonian Systems," Springer-Verlag, Berlin, 2000.
[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.
[10]	W. M. Zou and M. Schechter, "Critical Point Theory and its Applications," Springer, 2006.
[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.
[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.
[13]	W. Omana and M. Willem, Homoclinic orbits for a class of Hamiltonian systems, Diff. and Int. Eq., 5 (1992), 1115-1120.
[14]	P. H. Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems, Proc. Roy. Soc. Edinburgh, 114 (1990), 33-38.
[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.
[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.
[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.
[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.
[19]	D. E. Edmunds and W. D. Evans, "Spectral Theory and Differential Operators, Clarendon Press," Oxford, 1987.
[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.
[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.
[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.
[23]	E. Séré, Existence of infinitely many homoclinic orbits in Hamiltonian systems, Math. Z., 209 (1992), 27-42. doi: doi:10.1007/BF02570817.
[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.
[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.
[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.
[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.
[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.
[29]	M. Schechter and W. M. Zou, Homoclinic orbits for Schrödinger systems, Michigan Math. J., 51 (2005), 59-71.
[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.

show all references

References:

[1]	Y. H. Ding, "Variational Methods for Strongly Indefinite Problems," World Scientific Press, 2008.
[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.
[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.
[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.
[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.
[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.
[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.
[8]	M. Struwe, "Variational Methods Applications to Nolinear Partial Differential Equations and Hamiltonian Systems," Springer-Verlag, Berlin, 2000.
[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.
[10]	W. M. Zou and M. Schechter, "Critical Point Theory and its Applications," Springer, 2006.
[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.
[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.
[13]	W. Omana and M. Willem, Homoclinic orbits for a class of Hamiltonian systems, Diff. and Int. Eq., 5 (1992), 1115-1120.
[14]	P. H. Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems, Proc. Roy. Soc. Edinburgh, 114 (1990), 33-38.
[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.
[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.
[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.
[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.
[19]	D. E. Edmunds and W. D. Evans, "Spectral Theory and Differential Operators, Clarendon Press," Oxford, 1987.
[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.
[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.
[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.
[23]	E. Séré, Existence of infinitely many homoclinic orbits in Hamiltonian systems, Math. Z., 209 (1992), 27-42. doi: doi:10.1007/BF02570817.
[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.
[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.
[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.
[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.
[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.
[29]	M. Schechter and W. M. Zou, Homoclinic orbits for Schrödinger systems, Michigan Math. J., 51 (2005), 59-71.
[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.

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