# American Institute of Mathematical Sciences

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

 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.
Citation: Juntao Sun, Jifeng Chu, Zhaosheng Feng. Homoclinic orbits for first order periodic Hamiltonian systems with spectrum point zero. Discrete and 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [11] V. Coti Zelati and P. H. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on $\mathbb{R}^N2$, Comm. Pure Appl. Math., 45 (1992), 1217-1269. doi: 10.1002/cpa.3160451002. [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. [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. [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. [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. [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. [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. [18] Y. Ding and M. Willem, Homoclinic orbits of a Hamiltonian system, Z. Angew. Math. Phys., 50 (1999), 759-778. doi: 10.1007/s000330050177. [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. [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. [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. [22] J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems," Applied Mathematical Sciences, 74, Springer-Verlag, New York, 1989. [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. [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. [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. [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. [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. [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. [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. [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. [31] S. Zhang, Symmetrically homoclinic orbits for symmetric Hamiltonian systems, J. Math. Anal. Appl., 247 (2000), 645-652. doi: 10.1006/jmaa.2000.6839.

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##### 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [11] V. Coti Zelati and P. H. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on $\mathbb{R}^N2$, Comm. Pure Appl. Math., 45 (1992), 1217-1269. doi: 10.1002/cpa.3160451002. [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. [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. [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. [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. [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. [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. [18] Y. Ding and M. Willem, Homoclinic orbits of a Hamiltonian system, Z. Angew. Math. Phys., 50 (1999), 759-778. doi: 10.1007/s000330050177. [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. [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. [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. [22] J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems," Applied Mathematical Sciences, 74, Springer-Verlag, New York, 1989. [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. [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. [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. [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. [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. [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. [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. [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. [31] S. Zhang, Symmetrically homoclinic orbits for symmetric Hamiltonian systems, J. Math. Anal. Appl., 247 (2000), 645-652. doi: 10.1006/jmaa.2000.6839.
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