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An Index Theory with Applications to Homoclinic Orbits of Hamiltonian Systems and Dirac Equations

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Abstract

In this paper, we will define the index pair \((i_A(B),\nu _A(B))\) by the dual variational method, and show the relationship between the indices defined by different methods. As applications, we apply the index \((i_A(B),\nu _A(B))\) to study the existence and multiplicity of homoclinic orbits of nonlinear Hamiltonian systems and solutions of nonlinear Dirac equations.

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References

  1. Abbondandolo, A.: Morse Theory for Hamiltonian Systemsg. Chapman & Hall/CRC, London (2001)

    MATH  Google Scholar 

  2. Amann, H.: Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev. 18, 620–709 (1976)

    MathSciNet  MATH  Google Scholar 

  3. Amann, H., Zehnder, E.: Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations. Annali Scuola Norm. Sup. Pisa 7, 539–603 (1980)

    MathSciNet  MATH  Google Scholar 

  4. Arioli, G., Szulkin, A.: Homoclinic solutions of Hamiltonian systems with symmetry. J. Differ. Equ. 158, 291–313 (1999)

    MathSciNet  MATH  Google Scholar 

  5. Aubin, J.P., Ekeland, I.: Applied Nonlinear Analysis. Wiley, London (1984)

    MATH  Google Scholar 

  6. Bartsch, T., Ding, Y.: Solutions of nonlinear Dirac equations. J. Differ. Equ. 226, 210–249 (2006)

    MathSciNet  MATH  Google Scholar 

  7. Chang, K.Q.: Infinite Dimensional Morse Theory and Multiple Solution Problems. Birkhauser, Basel (1993)

    MATH  Google Scholar 

  8. Chang, K.C., Liu, J.Q., Liu, M.J.: Nontrivial periodic solutions for strong resonance Hamiltonian systems. Ann. Inst. H. PoincaršŠ Anal. Non LinšŠaire 14, 103–117 (1997)

    MathSciNet  MATH  Google Scholar 

  9. Chen, C., Hu, X.: Maslov index for homoclinic orbits of Hamiltonian systems. Ann. Inst. H. Poincaré, Anal. Non lin éaire 24, 589–603 (2007)

    MathSciNet  MATH  Google Scholar 

  10. Chen, G., Ma, S.: Homoclinic orbits of superlinear Hamiltonian systems. Proc. Am. Math. Soc. 139, 3973–3983 (2011)

    MathSciNet  MATH  Google Scholar 

  11. Clark, D.C.: A varant of Ljusternik–Schnirelman theory. Ind. Univ. Math. J. 22, 65–74 (1972)

    Google Scholar 

  12. Conley, C., Zehnder, E.: Morse-type index theory for flows and periodic solutions for Hamiltonian equations. Commun. Pure Appl. Math. 37, 207–253 (1984)

    MathSciNet  MATH  Google Scholar 

  13. Ding, Y.: Multiple homoclinics in a Hamiltonian system with asymptotically or super linear terms. Commun. Contemp. Math. 8, 453–480 (2006)

    MathSciNet  MATH  Google Scholar 

  14. Ding, Y.: Variational Methods for Strongly Indefinite Problems. World Scientific Publishing, Singapore (2007)

    MATH  Google Scholar 

  15. Ding, Y., Jeanjean, L.: Homoclinic orbits for a nonperiodic Hamiltonian system. J. Differ. Equ. 237, 473–490 (2007)

    MathSciNet  MATH  Google Scholar 

  16. Ding, Y., Girardi, M.: Infinitely many homoclinic orbits of a Hamiltonian system with symmetry. Nonlinear Anal. 38, 391–415 (1999)

    MathSciNet  MATH  Google Scholar 

  17. Ding, Y., Li, S.: Homoclinic orbits for first order Hamiltonian systems. J. Math. Anal. Appl. 189, 585–601 (1995)

    MathSciNet  MATH  Google Scholar 

  18. Ding, Y., Lee, C.: Existence and exponential decay of homoclinics in a nonperiodic superquadratic Hamiltonian system. J. Differ. Equ. 246, 2829–2848 (2009)

    MathSciNet  MATH  Google Scholar 

  19. Ding, Y., Liu, X.: On Semiclassical Ground States of a Nonlinear Dirac Equation. Interdisciplinary Mathematical Sciences, vol. 7. World Scientific, Singapore (2007)

    MATH  Google Scholar 

  20. Ding, Y., Ruf, B.: Solutions of a nonlinear Dirac equation with external fields. Arch. Ration. Mech. Anal. 190, 1007–1032 (2008)

    MathSciNet  MATH  Google Scholar 

  21. Ding, Y., Xu, T.: Localized concentration of semi-classical states for nonlinear Dirac equations. Arch. Ration. Mech. Anal. 216, 415–447 (2015)

    MathSciNet  MATH  Google Scholar 

  22. Ding, Y., Willem, M.: Homoclinic orbits of a Hamiltonian system. Z. Angew. Math. Phys. 50, 759–778 (1999)

    MathSciNet  MATH  Google Scholar 

  23. Dong, D., Long, Y.: The iteration formula of Maslov-type index theory with applications to nonlinear Hamiltonian systems. Trans. Am. Math. Soc. 349, 2619–2661 (1997)

    MathSciNet  MATH  Google Scholar 

  24. Dong, Y.: Index theory for linear selfadjoint operator equations and nontrivial solutions for asymptotically linear operator equations. Calc. Var. 38, 75–109 (2010)

    MathSciNet  MATH  Google Scholar 

  25. Ekeland, I.: Une theorie de Morse pour les systemes hamiltoniens convexes. Ann IHP Anal. non lineaire 1, 19–78 (1984)

    MathSciNet  MATH  Google Scholar 

  26. Ekeland, I.: Convexity Methods in Hamiltonian Mechanics. Springer, Berlin (1990)

    MATH  Google Scholar 

  27. Ekeland, I., Hofer, H.: Periodic solutions with prescribed period for convex autonomous Hamiltonian systems. Invent. Math. 81, 155–188 (1985)

    MathSciNet  MATH  Google Scholar 

  28. Ekeland, I., Hofer, H.: Convex Hamiltonian energy surfaces and their closed trajectories. Commun. Math. Phys. 113, 419–467 (1987)

    MATH  Google Scholar 

  29. Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. North-Holland-Elsevier, Amsterdam (1976)

    MATH  Google Scholar 

  30. Esteban, M., Séré, E.: Stationary states of the nonlinear Dirac equation: a variational approach. Commun. Math. Phys. 171, 323–350 (1995)

    MathSciNet  MATH  Google Scholar 

  31. Fei, G.: Relative Morse index and its application to Hamiltonian systems in the Presence of symmetries. J. Differ. Equ. 122, 302–315 (1995)

    MathSciNet  MATH  Google Scholar 

  32. Figueiredo, G.M., Pimenta, M.T.O.: Existence of ground state solutions to Dirac equations with vanishing potentials at infinity. J. Differ. Equ. 262, 486–505 (2017)

    MathSciNet  MATH  Google Scholar 

  33. Hofer, H., Wysocki, K.: First order elliptic systems and the existence of homoclinic orbits in Hamiltonian systems. Math. Ann. 228, 483–503 (1990)

    MathSciNet  MATH  Google Scholar 

  34. Liu, C.: Maslov-type index theory for symplectic paths with Lagrangian boundary conditions. Adv. Nonlinear Stud. 7, 131–161 (2007)

    MathSciNet  MATH  Google Scholar 

  35. Liu, C.: Asymptotically linear Hamiltonian system with Lagrangian boundary conditions. Pac. J. Math. 232, 232–254 (2007)

    MathSciNet  Google Scholar 

  36. Liu, C., Long, Y., Zhu, C.: Multiplicity of closed characteristics on symmetric convex hypersurfaces in \(\mathbb{R}^{2n}\). Math. Ann. 323, 201–215 (2002)

    MathSciNet  MATH  Google Scholar 

  37. Liu, C., Wang, Q., Lin, X.: An index theory for symplectic paths associated with two Lagrangian subspaces with applications. Nonlinearity 24, 43–70 (2011)

    MathSciNet  MATH  Google Scholar 

  38. Long, Y.: Maslov-type index, degenerate critical points, and asymptotically linear Hamiltonian systems. Sci. China 33, 1409–1419 (1990)

    MathSciNet  MATH  Google Scholar 

  39. Long, Y.: A Maslov-type index theory for symplectic paths. Topol. Methods Nonlinear Anal. 10, 47–78 (1997)

    MathSciNet  MATH  Google Scholar 

  40. Long, Y., Zehnder, E.: Morse theory for forced oscillations of asymptotically linear Hamiltonian systems. In: Alberverio, S., et al. (eds.) Stochastic Processes, Physics and Geometry, pp. 528–563. World Scientific, Teaneck, NJ (1990)

    Google Scholar 

  41. Zhu, C., Long, Y.: Maslov type index theory for symplectiuc paths and spectral flow(I). Chin. Ann. Math. 20B, 413–424 (1999)

    MATH  Google Scholar 

  42. Long, Y., Zhu, C.: Maslov type index theory for symplectiuc paths and spectral flow(II). Chin. Ann. Math. 21B, 89–108 (2000)

    MATH  Google Scholar 

  43. Long, Y., Zhu, C.: Closed characteristics on compact convex hypersurfaces in \(\mathbb{R}^{2n}\). Ann. Math. 155, 317–368 (2000)

    MATH  Google Scholar 

  44. Merle, F.: Existence of stationary states for Dirac equations. J. Differ. Equ. 74, 50–68 (1988)

    MathSciNet  MATH  Google Scholar 

  45. Séré, E.: Existence of infinitely many homoclinic orbits in Hamiltonian systems. Math. Z. 209, 27–42 (1992)

    MathSciNet  MATH  Google Scholar 

  46. Séré, E.: Looking for the Bernoulli shift. Ann. Inst. H. Poincaré, Anal. Non linéaire 10, 561–590 (1993)

    MathSciNet  MATH  Google Scholar 

  47. Sun, J., Chen, H., Nieto, J.: Homoclinic orbits for a class of first-order nonperiodic asymptotically quadratic Hamiltonian systems with spectrum point zero. J. Math. Anal. Appl. 378, 117–127 (2011)

    MathSciNet  MATH  Google Scholar 

  48. Sun, J., Chu, J., Feng, Z.: Homoclinic orbits for first order periodic Hamiltonian systems with spectrum point zero. Discrete Contin. Dyn. Syst. 33, 3807–3824 (2013)

    MathSciNet  MATH  Google Scholar 

  49. Szulkin, A., Zou, W.: Homoclinic orbits for asymptotically linear Hamiltonian systems. J. Funct. Anal. 187, 25–41 (2001)

    MathSciNet  MATH  Google Scholar 

  50. Tanaka, K.: Homoclinic orbits in a first order superquadratic Hamiltonian system: convergence of subharmonic orbits. J. Differ. Equ. 94, 315–339 (1991)

    MathSciNet  MATH  Google Scholar 

  51. Wang, J., Xu, J., Zhang, F.: Homoclinic orbits of superlinear Hamiltonian systems without Ambrosetti–Rabinowitz growth condition. Discrete Contin. Dyn. Syst. 27, 1241–1257 (2010)

    MathSciNet  MATH  Google Scholar 

  52. Wang, J., Xu, J., Zhang, F.: Infinitely many homoclinic orbits for superlinear Hamiltonian systems. Topol. Methods Nonlinear Anal. 39, 1–22 (2012)

    MathSciNet  MATH  Google Scholar 

  53. Wang, Q., Liu, C.: Periodic solutions of delay differential systems via Hamiltonian systems. Nonlinear Ana. TMA 102, 159–167 (2014)

    MathSciNet  MATH  Google Scholar 

  54. Wang, Q., Liu, C.: The relative Morse index theory for infinite dimensional Hamiltonian systems with applications. J. Math. Anal. Appl. 427, 17–30 (2015)

    MathSciNet  MATH  Google Scholar 

  55. Wang, Q., Liu, C.: A new index theory for linear self-adjoint operator equations and its applications. J. Differ. Equ. 260, 3749–3784 (2016)

    MathSciNet  MATH  Google Scholar 

  56. Zelati, V.C., Ekeland, I., Séré, E.: A variational approach to homoclinic orbits in Hamiltonian systems. Math. Ann. 228, 133–160 (1990)

    MathSciNet  MATH  Google Scholar 

  57. Zhang, J., Tang, X., Zhang, W.: Homoclinic orbits of nonperiodic superquadratic Hamiltonian system. Taiwan. J. Math. 17, 1855–1867 (2013)

    MathSciNet  MATH  Google Scholar 

  58. Zhang, W., Tang, X., Zhang, J.: Homoclinlic solutions for the first-order Hamiltonian system with superquadratic nonlinearity. Taiwan. J. Math. 19, 673–690 (2015)

    MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Statistics, Henan University, Kaifeng, 475000, People’s Republic of China

    Qi Wang

  2. School of Mathematics and Information Science, Guangzhou University, Guangzhou, 510006, People’s Republic of China

    Chungen Liu

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  1. Qi Wang
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  2. Chungen Liu
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Correspondence to Chungen Liu.

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Qi Wang: Partially supported by NNSF of China (11301148). Chungen Liu: Partially supported by NNSF of China (11790271), Guangdong Basic and Applied basic Research Foundation (2020A1515011019), Innovation and development project of Guangzhou University, and Nankai Zhide Foundation.

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Wang, Q., Liu, C. An Index Theory with Applications to Homoclinic Orbits of Hamiltonian Systems and Dirac Equations. J Dyn Diff Equat 32, 1177–1201 (2020). https://doi.org/10.1007/s10884-020-09846-3

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  • DOI: https://doi.org/10.1007/s10884-020-09846-3

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