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

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.

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:
[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.  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.  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.  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.  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.   Google Scholar

[7]

Y. H. Ding and M. Girardi, Infinitely many homoclinic orbits of a Hamiltonian system with symmetry,, Nonlinear Anal., 38 (1999), 391.  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, (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., (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.  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.  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.   Google Scholar

[14]

P. H. Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems,, Proc. Roy. Soc. Edinburgh, 114 (1990), 33.   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.  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.  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.  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.  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.  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.  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.  doi: doi:10.1007/BF01444543.  Google Scholar

[23]

E. Séré, Existence of infinitely many homoclinic orbits in Hamiltonian systems,, Math. Z., 209 (1992), 27.  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.  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.  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.  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.  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.  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.   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.  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.  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.  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.  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.  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.   Google Scholar

[7]

Y. H. Ding and M. Girardi, Infinitely many homoclinic orbits of a Hamiltonian system with symmetry,, Nonlinear Anal., 38 (1999), 391.  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, (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., (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.  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.  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.   Google Scholar

[14]

P. H. Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems,, Proc. Roy. Soc. Edinburgh, 114 (1990), 33.   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.  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.  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.  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.  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.  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.  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.  doi: doi:10.1007/BF01444543.  Google Scholar

[23]

E. Séré, Existence of infinitely many homoclinic orbits in Hamiltonian systems,, Math. Z., 209 (1992), 27.  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.  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.  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.  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.  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.  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.   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.  doi: doi:10.1142/S0219199706002192.  Google Scholar

[1]

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

[2]

Xing-Bin Pan. Variational and operator methods for Maxwell-Stokes system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3909-3955. doi: 10.3934/dcds.2020036

[3]

Ivan Bailera, Joaquim Borges, Josep Rifà. On Hadamard full propelinear codes with associated group $ C_{2t}\times C_2 $. Advances in Mathematics of Communications, 2021, 15 (1) : 35-54. doi: 10.3934/amc.2020041

[4]

Jan Bouwe van den Berg, Elena Queirolo. A general framework for validated continuation of periodic orbits in systems of polynomial ODEs. Journal of Computational Dynamics, 2021, 8 (1) : 59-97. doi: 10.3934/jcd.2021004

[5]

Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024

[6]

Sihem Guerarra. Maximum and minimum ranks and inertias of the Hermitian parts of the least rank solution of the matrix equation AXB = C. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 75-86. doi: 10.3934/naco.2020016

[7]

Christian Aarset, Christian Pötzsche. Bifurcations in periodic integrodifference equations in $ C(\Omega) $ I: Analytical results and applications. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 1-60. doi: 10.3934/dcdsb.2020231

[8]

Kengo Matsumoto. $ C^* $-algebras associated with asymptotic equivalence relations defined by hyperbolic toral automorphisms. Electronic Research Archive, , () : -. doi: 10.3934/era.2021006

[9]

Simon Hochgerner. Symmetry actuated closed-loop Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 641-669. doi: 10.3934/jgm.2020030

[10]

Hua Shi, Xiang Zhang, Yuyan Zhang. Complex planar Hamiltonian systems: Linearization and dynamics. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020406

[11]

Manuel de León, Víctor M. Jiménez, Manuel Lainz. Contact Hamiltonian and Lagrangian systems with nonholonomic constraints. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2021001

[12]

Waixiang Cao, Lueling Jia, Zhimin Zhang. A $ C^1 $ Petrov-Galerkin method and Gauss collocation method for 1D general elliptic problems and superconvergence. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 81-105. doi: 10.3934/dcdsb.2020327

[13]

Beom-Seok Han, Kyeong-Hun Kim, Daehan Park. A weighted Sobolev space theory for the diffusion-wave equations with time-fractional derivatives on $ C^{1} $ domains. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021002

[14]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138

[15]

Adrian Viorel, Cristian D. Alecsa, Titus O. Pinţa. Asymptotic analysis of a structure-preserving integrator for damped Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020407

[16]

Huanhuan Tian, Maoan Han. Limit cycle bifurcations of piecewise smooth near-Hamiltonian systems with a switching curve. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020368

[17]

Thierry Horsin, Mohamed Ali Jendoubi. On the convergence to equilibria of a sequence defined by an implicit scheme. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020465

[18]

Andrew D. Lewis. Erratum for "nonholonomic and constrained variational mechanics". Journal of Geometric Mechanics, 2020, 12 (4) : 671-675. doi: 10.3934/jgm.2020033

[19]

Héctor Barge. Čech cohomology, homoclinic trajectories and robustness of non-saddle sets. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020381

[20]

Constantine M. Dafermos. A variational approach to the Riemann problem for hyperbolic conservation laws. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 185-195. doi: 10.3934/dcds.2009.23.185

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (41)
  • HTML views (0)
  • Cited by (22)

Other articles
by authors

[Back to Top]