American Institute of Mathematical Sciences

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August  2013, 33(8): 3807-3824. doi: 10.3934/dcds.2013.33.3807

Homoclinic orbits for first order periodic Hamiltonian systems with spectrum point zero

Juntao Sun 1, , Jifeng Chu 2, and  Zhaosheng Feng 3,

1. 

School of Science, Shandong University of Technology, Zibo 255049, China
2. 

Department of Mathematics, College of Science, Hohai University, Nanjing 210098, China
3. 

Department of Mathematics, University of Texas-Pan American, Edinburg, TX 78539

Received  September 2012 Revised  November 2012 Published  January 2013

In this paper, we study the existence and multiplicity of homoclinic orbits for a class of first order periodic Hamiltonian systems. By applying two recent critical point theorems for strongly indefinite functionals, we establish some new criteria to guarantee that Hamiltonian systems, with asymptotically quadratic terms and spectrum point zero, have at least one and infinitely many homoclinic orbits under certain conditions.
Keywords: variational methods, Hölder inequality, Strongly indefinite problems., Homoclinic orbits, Hamiltonian systems, asymptotically linear terms.
Mathematics Subject Classification: Primary: 58E05, 70H05; Secondary: 37J4.
Citation: Juntao Sun, Jifeng Chu, Zhaosheng Feng. Homoclinic orbits for first order periodic Hamiltonian systems with spectrum point zero. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3807-3824. doi: 10.3934/dcds.2013.33.3807
References:
[1]

N. Ackermann, On a periodic Schrödinger equation with nonlocal superlinear part, Math. Z., 248 (2004), 423-443. doi: 10.1007/s00209-004-0663-y.  Google Scholar

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N. Ackermann, A nonlinear superposition principle and multibump solutions of periodic Schrödinger equations, J. Funct. Anal., 234 (2006), 277-320. doi: 10.1016/j.jfa.2005.11.010.  Google Scholar

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G. Arioli and A. Szulkin, Homoclinic solutions of Hamiltonian systems with symmetry, J. Differential Equations, 158 (1999), 291-313. doi: 10.1006/jdeq.1999.3639.  Google Scholar

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T. Bartsch and Y. Ding, Deformation theorems on non-metrizable vector spaces and applications to critical point theory, Math. Nachr., 279 (2006), 1267-1288. doi: 10.1002/mana.200410420.  Google Scholar

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T. Bartsch and A. Szulkin, Hamiltonian Systems: Periodic and Homoclinic Solutions by Variational Methods, in "Handbook of Differential Equations: Ordinary Differential Equations," Vol. II, Elsevier B. V., Amsterdam, (2005), 77-146.  Google Scholar

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J. F. Chu, J. Z. Lei and M. R. Zhang, Lyapunov stability for conservative systems with lower degrees of freedom, Discrete Contin. Dyn. Syst. Ser. B, 16 (2011), 423-443. doi: 10.3934/dcdsb.2011.16.423.  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: 10.1007/BF01444526.  Google Scholar

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V. Coti Zelati and P. H. Rabinowitz, Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials, J. Amer. Math. Soc., 4 (1991), 693-727. doi: 10.2307/2939286.  Google Scholar

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V. Coti Zelati and P. H. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on $\mathbbR^N$, Comm. Pure Appl. Math., 45 (1992), 1217-1269. doi: 10.1002/cpa.3160451002.  Google Scholar

[12]

Y. Ding, "Variational Methods for Strongly Indefinite Problems," Interdiscip. Math. Sci., 7, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007. doi: 10.1142/9789812709639.  Google Scholar

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Y. Ding, Multiple homoclinics in a Hamiltonian system with asymptotically or super linear terms, Commun. Contemp. Math., 8 (2006), 453-480. doi: 10.1142/S0219199706002192.  Google Scholar

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Y. 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.  Google Scholar

[15]

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

[16]

Y. Ding and C. Lee, Existence and exponential decay of homoclinics in a nonperiodic superquadratic Hamiltonian system, J. Differential Equations, 246 (2009), 2829-2848. doi: 10.1016/j.jde.2008.12.013.  Google Scholar

[17]

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

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

[19]

Z. Feng and D. Y. Gao, An asymptotic expression of the Schrödinger equation, Z. Angew. Math. Phys., 60 (2009), 363-375. doi: 10.1007/s00033-007-7102-y.  Google Scholar

[20]

Z. Feng and Y. Huang, Approximate solution of the Burgers-Korteweg-de Vries equation, Commun. Pure Appl. Anal., 6 (2007), 429-440. doi: 10.3934/cpaa.2007.6.429.  Google Scholar

[21]

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

[22]

J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems," Applied Mathematical Sciences, 74, Springer-Verlag, New York, 1989.  Google Scholar

[23]

A. Mielke, Weak-convergence methods for hamiltonian multiscale problems, Discrete Contin. Dyn. Syst., 20 (2008), 53-79. doi: 10.3934/dcds.2008.20.53.  Google Scholar

[24]

O. Koltsova and L. Lerman, Hamiltonian dynamics near nontransverse homoclinic orbit to saddle-focus equilibrium, Discrete Contin. Dyn. Syst., 25 (2009), 883-913. doi: 10.3934/dcds.2009.25.883.  Google Scholar

[25]

I. SenGupta and M. C. Mariani, Spherical harmonics applied to differential and integro-differential equations arising in mathematical finance, Differ. Equ. Dyn. Syst., 20 (2012), 93-109. doi: 10.1007/s12591-012-0107-9.  Google Scholar

[26]

J. Sun, H. Chen, J. J. Nieto and M. Otero-Novoa, Multiplicity of solutions for perturbed second-order Hamiltonian systems with impulsive effects, Nonlinear Anal., 72 (2010), 4575-4586. doi: 10.1016/j.na.2010.02.034.  Google Scholar

[27]

J. Sun, H. Chen and J. J. Nieto, Homoclinic solutions for a class of subquadratic second-order Hamiltonian systems, J. Math. Anal. Appl., 373 (2011), 20-29. doi: 10.1016/j.jmaa.2010.06.038.  Google Scholar

[28]

J. Sun, H. Chen and J. J. Nieto, Homoclinic orbits for a class of first-order nonperiodic asymptotically quadratic Hamiltonian systems with spectrum point zero, J. Math. Anal. Appl., 378 (2011), 117-127. doi: 10.1016/j.jmaa.2010.12.044.  Google Scholar

[29]

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.  Google Scholar

[30]

J. Wang, J. Xu and F. Zhang, Homoclinic orbits of superlinear Hamiltonian systems without Ambrosetti-Rabinowitz growth condition, Discrete Contin. Dyn. Syst., 27 (2010), 1241-1257. doi: 10.3934/dcds.2010.27.1241.  Google Scholar

[31]

S. Zhang, Symmetrically homoclinic orbits for symmetric Hamiltonian systems, J. Math. Anal. Appl., 247 (2000), 645-652. doi: 10.1006/jmaa.2000.6839.  Google Scholar

show all references

References:
[1]

N. Ackermann, On a periodic Schrödinger equation with nonlocal superlinear part, Math. Z., 248 (2004), 423-443. doi: 10.1007/s00209-004-0663-y.  Google Scholar

[2]

N. Ackermann, A nonlinear superposition principle and multibump solutions of periodic Schrödinger equations, J. Funct. Anal., 234 (2006), 277-320. doi: 10.1016/j.jfa.2005.11.010.  Google Scholar

[3]

G. Arioli and A. Szulkin, Homoclinic solutions of Hamiltonian systems with symmetry, J. Differential Equations, 158 (1999), 291-313. doi: 10.1006/jdeq.1999.3639.  Google Scholar

[4]

T. Bartsch and Y. Ding, Deformation theorems on non-metrizable vector spaces and applications to critical point theory, Math. Nachr., 279 (2006), 1267-1288. doi: 10.1002/mana.200410420.  Google Scholar

[5]

T. Bartsch and A. Szulkin, Hamiltonian Systems: Periodic and Homoclinic Solutions by Variational Methods, in "Handbook of Differential Equations: Ordinary Differential Equations," Vol. II, Elsevier B. V., Amsterdam, (2005), 77-146.  Google Scholar

[6]

G. Bonanno and R. Livrea, Multiple periodic solutions for Hamiltonian systems with not coercive potential, J. Math. Anal. Appl., 363 (2010), 627-638. doi: 10.1016/j.jmaa.2009.09.025.  Google Scholar

[7]

C. Chen and X. Hu, Maslov index for homoclinic orbits of Hamiltonian systems, Ann. I. H. Poincaré Anal. Linéaire, 24 (2007), 589-603. doi: 10.1016/j.anihpc.2006.06.002.  Google Scholar

[8]

J. F. Chu, J. Z. Lei and M. R. Zhang, Lyapunov stability for conservative systems with lower degrees of freedom, Discrete Contin. Dyn. Syst. Ser. B, 16 (2011), 423-443. doi: 10.3934/dcdsb.2011.16.423.  Google Scholar

[9]

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.  Google Scholar

[10]

V. Coti Zelati and P. H. Rabinowitz, Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials, J. Amer. Math. Soc., 4 (1991), 693-727. doi: 10.2307/2939286.  Google Scholar

[11]

V. Coti Zelati and P. H. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on $\mathbbR^N$, Comm. Pure Appl. Math., 45 (1992), 1217-1269. doi: 10.1002/cpa.3160451002.  Google Scholar

[12]

Y. Ding, "Variational Methods for Strongly Indefinite Problems," Interdiscip. Math. Sci., 7, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007. doi: 10.1142/9789812709639.  Google Scholar

[13]

Y. Ding, Multiple homoclinics in a Hamiltonian system with asymptotically or super linear terms, Commun. Contemp. Math., 8 (2006), 453-480. doi: 10.1142/S0219199706002192.  Google Scholar

[14]

Y. 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.  Google Scholar

[15]

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

[16]

Y. Ding and C. Lee, Existence and exponential decay of homoclinics in a nonperiodic superquadratic Hamiltonian system, J. Differential Equations, 246 (2009), 2829-2848. doi: 10.1016/j.jde.2008.12.013.  Google Scholar

[17]

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

[18]

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

[19]

Z. Feng and D. Y. Gao, An asymptotic expression of the Schrödinger equation, Z. Angew. Math. Phys., 60 (2009), 363-375. doi: 10.1007/s00033-007-7102-y.  Google Scholar

[20]

Z. Feng and Y. Huang, Approximate solution of the Burgers-Korteweg-de Vries equation, Commun. Pure Appl. Anal., 6 (2007), 429-440. doi: 10.3934/cpaa.2007.6.429.  Google Scholar

[21]

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

[22]

J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems," Applied Mathematical Sciences, 74, Springer-Verlag, New York, 1989.  Google Scholar

[23]

A. Mielke, Weak-convergence methods for hamiltonian multiscale problems, Discrete Contin. Dyn. Syst., 20 (2008), 53-79. doi: 10.3934/dcds.2008.20.53.  Google Scholar

[24]

O. Koltsova and L. Lerman, Hamiltonian dynamics near nontransverse homoclinic orbit to saddle-focus equilibrium, Discrete Contin. Dyn. Syst., 25 (2009), 883-913. doi: 10.3934/dcds.2009.25.883.  Google Scholar

[25]

I. SenGupta and M. C. Mariani, Spherical harmonics applied to differential and integro-differential equations arising in mathematical finance, Differ. Equ. Dyn. Syst., 20 (2012), 93-109. doi: 10.1007/s12591-012-0107-9.  Google Scholar

[26]

J. Sun, H. Chen, J. J. Nieto and M. Otero-Novoa, Multiplicity of solutions for perturbed second-order Hamiltonian systems with impulsive effects, Nonlinear Anal., 72 (2010), 4575-4586. doi: 10.1016/j.na.2010.02.034.  Google Scholar

[27]

J. Sun, H. Chen and J. J. Nieto, Homoclinic solutions for a class of subquadratic second-order Hamiltonian systems, J. Math. Anal. Appl., 373 (2011), 20-29. doi: 10.1016/j.jmaa.2010.06.038.  Google Scholar

[28]

J. Sun, H. Chen and J. J. Nieto, Homoclinic orbits for a class of first-order nonperiodic asymptotically quadratic Hamiltonian systems with spectrum point zero, J. Math. Anal. Appl., 378 (2011), 117-127. doi: 10.1016/j.jmaa.2010.12.044.  Google Scholar

[29]

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.  Google Scholar

[30]

J. Wang, J. Xu and F. Zhang, Homoclinic orbits of superlinear Hamiltonian systems without Ambrosetti-Rabinowitz growth condition, Discrete Contin. Dyn. Syst., 27 (2010), 1241-1257. doi: 10.3934/dcds.2010.27.1241.  Google Scholar

[31]

S. Zhang, Symmetrically homoclinic orbits for symmetric Hamiltonian systems, J. Math. Anal. Appl., 247 (2000), 645-652. doi: 10.1006/jmaa.2000.6839.  Google Scholar

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