Skip to main content
SpringerLink
Log in

Non-Nehari manifold method for periodic discrete superlinear Schrödinger equation

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

We consider the nonlinear difference equations of the form

$$Lu = f\left( {n,u} \right),\;n \in Z,$$

where L is a Jacobi operator given by (Lu)(n) = a(n)u(n+1)+a(n−1)u(n−1)+b(n)u(n) for n ∈ Z, {a(n)} and {b(n)} are real valued N-periodic sequences, and f(n, t) is superlinear on t. Inspired by previous work of Pankov [Discrete Contin. Dyn. Syst., 19, 419–430 (2007)] and Szulkin and Weth [J. Funct. Anal., 257, 3802–3822 (2009)], we develop a non-Nehari manifold method to find ground state solutions of Nehari–Pankov type under weaker conditions on f. Unlike the Nehari manifold method, the main idea of our approach lies on finding a minimizing Cerami sequence for the energy functional outside the Nehari–Pankov manifold by using the diagonal method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aceves, A. B.: Optical gap solutions: Past, persent, and future; theory and experiments. Chaos, 10, 584–589 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bronski, J. C., Segev, M., Weinstein, M. I.: Mathematical frontiers in optical solitons. Proc. Nat. Acad. Sci. USA, 98, 12872–12873 (2001)

    Article  MathSciNet  Google Scholar 

  3. Chen, G., Ma, S.: Discrete nonlinear Schrödinger equations with superlinear nonlinearities. Appl. Math. Comput., 218, 5496–5507 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ding, Y.: Variational Methods for Strongly Indefinite Problems, World Scientific, Singapore, 2007

    MATH  Google Scholar 

  5. Edmunds, D. E., Evans, W. D.: Spectral Theory and Differential Operators, Clarendon Press, Oxford, 1987

    MATH  Google Scholar 

  6. de Sterke, C. M., Sipe, J. E.: Gap Solitons, Progress in Optics, 33, (Ed. E. Wolf), North-Holland, Amsterdam, 1994, 203–260

    Google Scholar 

  7. Flash, S., Willis, C. R.: Discrete breathers. Phys. Repts., 295, 181–264 (1998)

    Article  MathSciNet  Google Scholar 

  8. Fleischer, J. W., Carmon, T., Segev, M., et al.: Observation of discrete solitons in optically-induced realtime waveguide arrows. Phys. Rew. Letts., 90, 023902 (2003)

    Article  Google Scholar 

  9. Hennig, D., Tsironis, G. P.: Wave transmission in nonlinear lattices. Physics Repts., 309 333–432 (1999)

    Article  MathSciNet  Google Scholar 

  10. Kevrekides, P. G., Rasmussen, K., Bishop, A. R.: The discrete nonlinear Schrödinger equation: a survey of recent results. Intern. J. Modern Phys. B, 15, 2833–2900 (2001)

    Article  Google Scholar 

  11. Kryszewski, W., Szulkin, A.: Generalized linking theorem with an application to a semilinear Schrödinger equation. Adv. Differ. Equ., 3, 441–472 (1998)

    MathSciNet  MATH  Google Scholar 

  12. Li, G. B., Szulkin, A.: An asymptotically periodic Schrödinger equation with indefinite linear part. Commun. Contemp. Math., 4, 763–776 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lin, X. Y., Tang, X. H.: Existence of infinitely many homoclinic orbits in discrete Hamiltonian systems. J. Math. Anal. Appl., 373, 59–72 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Liu, S.: On superlinear Schrödinger equations with periodic potential. Calc. Var. Partial Differential Equations, 45, 1–9 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Mai, A., Zhou, Z.: Ground state solutions for the periodic discrete nonlinear Schrödinger equations with superlinear nonlinearities. Abst. Appl. Anal., 2013, Article ID317139, 1–11 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Mai, A., Zhou, Z.: Discrete solitons for periodic discrete nonlinear Schrödinger equations. Appl. Math. Comput., 222, 34–41 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Mills, D. L.: Nonlinear Optics. Basic Concepts, Springer, Berlin, 1998

    Book  MATH  Google Scholar 

  18. Pankov, A.: Periodic nonlinear Schrödinger equation with application to photonic crystals. Milan J. Math., 73, 259–287 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  19. Pankov, A.: Gap solitons in periodic discrete nonlinear Schrödinger equations. Nonlinearity, 19, 27–40 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  20. Pankov, A.: Travelling Waves and Periodic Oscillations in Fermi–Pasta–Ulam Latticies, Imperial College Press, London, 2005

    Book  MATH  Google Scholar 

  21. Pankov, A.: Gap solitons in periodic discrete nonlinear Schrödinger equations with saturable nonlinearities. J. Math. Anal. Appl., 371, 254–265 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Pankov, A., Pflüger, K.: Periodic and solitary travelling waves for the generalized Kkadontsev–Petviashvili equations. Math. Meth. Appl. Sci., 22, 733–752 (1999)

    Article  MATH  Google Scholar 

  23. Pankov, A., Pflüger, K.: On ground travelling waves for the generalized Kadomtsev–Petviashvili equations. Math. Phys., Anal., Geom., 3, 33–47 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  24. Pankov, A., Zakharchenko, N.: On some discrete variational problems. Acta Appl. Math., 65, 295–303 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  25. Pankov, A.: Gap solitons in periodic discrete nonlinear Schrödinger equations II: A generalized Nehari manifold approach. Discrete Contin. Dyn. Syst., 19, 419–430 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  26. Shi, H., Zhang, H.: Existence of gap solitons in periodic discrete nonlinear Schrödinger equations. J. Math. Anal. Appl., 361, 411–419 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Shi, H.: Gap solitons in periodic discrete Schrödinger equations with nonlinearity. Acta Appl. Math., 109, 1065–1075 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  28. Schechter, M.: Superlinear Schrödinger operators. J. Funct. Anal., 262, 2677–2694 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  29. Szulkin, A., Weth, T.: Ground state solutions for some indefinite variational problems. J. Funct. Anal., 257(12), 3802–3822 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  30. Tang, X. H.: Non-Nehari manifold method for superlinear Schrödinger equation. Taiwanese J. Math., 18, 1957–1979 (2014)

    MathSciNet  Google Scholar 

  31. Tang, X. H., Chen, J.: Infinitely many homoclinic orbits for a class of discrete Hamiltonian systems. Advances in Difference Equations, 2013:242 doi:10.1186/1687-1847-2013-242 (2013)

    Google Scholar 

  32. Tang, X. H., Lin, X. Y., Xiao, L.: Homoclinic solutions for a class of second order discrete Hamiltonian systems. J. Differ. Equ. Appl., 16, 1257–1273 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  33. Tang, X. H., Lin, X. Y.: Homoclinic solutions for a class of second order discrete Hamiltonian systems. Acta Math. Sin., Engl. Ser., 28, 609–622 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  34. Tang, X. H., Lin, X. Y.: Infinitely many homoclinic orbits for discrete Hamiltonian systems with subquadratic potential. J. Differ. Equ. Appl. 19, 796–813 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  35. Tang, X. H.: Non-Nehari manifold method for asymptotically periodic Schrödinger equations. Sci. China Math., 58, 715–728 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  36. Tang, X. H.: Non-Nehari manifold method for asymptotically linear Schrödinger equation. J. Aust. Math. Soc., 98, 104–116 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  37. Teschl, G.: Jacobi Operators and Completely Integrable Nonlinear Lattices, Math. Surveys Monogr., vol. 72, American Mathematical Society, Providence, RI, 2000

    MATH  Google Scholar 

  38. Weinstein, M. I.: Excitation thresholds for nonlinear localized modes on lattices. Nonlinearity, 12, 673–691 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  39. Yang, M., Chen, W., Ding, Y.: Solutions for discrete periodic Schrödinger equations with spectrum 0. Acta Appl. Math., 110, 1475–1488 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  40. Zhou, Z., Yu, J.: On the existence of homoclinic solutions of a class of discrete nonlinear periodic systems. J. Differ. Equations, 249, 1199–1212 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  41. Zhou, Z., Yu, J.: Homoclinic solutions in periodic nonlinear difference equations with superlinear nonlinearity. Acta Math. Sin., Engl. Ser., 29, 1809–1822 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  42. Zhou, Z., Yu, J., Chen, Y.: On the existence of gap solitons in a periodic discrete nonlinear Schrödinger equation with saturable nonlinearity. Nonlinearity, 23, 1727–1740 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  43. Zhou, Z., Yu, J. S., Chen, Y.: Homoclinic solutions in periodic difference equations with saturable nonlinearity. Sci. China Math., 54, 83–93 (2011)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Central South University, Changsha, 410083, P. R. China

    Xian Hua Tang

Authors
  1. Xian Hua Tang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xian Hua Tang.

Additional information

Supported by NSFC (Grant No. 11571370) and the Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20120162110021) of China

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tang, X.H. Non-Nehari manifold method for periodic discrete superlinear Schrödinger equation. Acta. Math. Sin.-English Ser. 32, 463–473 (2016). https://doi.org/10.1007/s10114-016-4262-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-016-4262-8

Keywords

MR(2010) Subject Classification

Search

Navigation