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Abstract and Applied Analysis
Volume 2012, Article ID 703596, 15 pages
http://dx.doi.org/10.1155/2012/703596
Research Article

Existence and Multiplicity Results of Homoclinic Solutions for the DNLS Equations with Unbounded Potentials

Defang Ma1,2 and Zhan Zhou1,2

1School of Mathematics and Information Science, Guangzhou University, Guangdong, Guangzhou 510006, China
2Key Laboratory of Mathematics and Interdisciplinary Sciences of Guangdong Higher Education Institutes, Guangzhou University, Guangdong, Guangzhou 510006, China

Received 25 July 2012; Accepted 9 September 2012

Academic Editor: Wenming Zou

Copyright © 2012 Defang Ma and Zhan Zhou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. G. Teschl, Jacobi Operators and Completely Integrable Nonlinear Lattices, vol. 72 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, USA, 2000.
  2. S. Flach and A. Gorbach, “Discrete breakers-Advances in theory and applications,” Physics Reports, vol. 467, pp. 1–116, 2008. View at Google Scholar
  3. S. Flach and C. R. Willis, “Discrete breathers,” Physics Reports, vol. 295, no. 5, pp. 181–264, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. D. Hennig and G. P. Tsironis, “Wave transmission in nonlinear lattices,” Physics Reports, vol. 307, no. 5-6, pp. 333–432, 1999. View at Publisher · View at Google Scholar
  5. G. Chen and S. Ma, “Discrete nonlinear Schrödinger equations with superlinear nonlinearities,” Applied Mathematics and Computation, vol. 218, no. 9, pp. 5496–5507, 2012. View at Publisher · View at Google Scholar
  6. A. Pankov, “Gap solitons in periodic discrete nonlinear Schrödinger equations,” Nonlinearity, vol. 19, no. 1, pp. 27–40, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. A. Pankov, “Standing waves for discrete non-linear Schrödinger equations: sign-changing nonlinearities,” Applicable Analysis. In press.
  8. G. Zhang and A. Pankov, “Standing wave solutions of the discrete non-linear Schrödinger equations with unbounded potentials—II,” Applicable Analysis, vol. 89, no. 9, pp. 1541–1557, 2010. View at Publisher · View at Google Scholar
  9. Z. Zhou, J. Yu, and Y. Chen, “Homoclinic solutions in periodic difference equations with saturable nonlinearity,” Science China Mathematics, vol. 54, no. 1, pp. 83–93, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. A. Pankov, “Gap solitons in periodic discrete nonlinear Schrödinger equations—II. A generalized Nehari manifold approach,” Discrete and Continuous Dynamical Systems. Series A, vol. 19, no. 2, pp. 419–430, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. A. Pankov, “Gap solitons in periodic discrete nonlinear Schrödinger equations with saturable nonlinearities,” Journal of Mathematical Analysis and Applications, vol. 371, no. 1, pp. 254–265, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. A. Pankov and V. Rothos, “Periodic and decaying solutions in discrete nonlinear Schrödinger with saturable nonlinearity,” Proceedings of The Royal Society of London. Series A, vol. 464, no. 2100, pp. 3219–3236, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. Z. Zhou and J. Yu, “On the existence of homoclinic solutions of a class of discrete nonlinear periodic systems,” Journal of Differential Equations, vol. 249, no. 5, pp. 1199–1212, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. Z. Zhou, J. Yu, and Y. Chen, “On the existence of gap solitons in a periodic discrete nonlinear Schrödinger equation with saturable nonlinearity,” Nonlinearity, vol. 23, no. 7, pp. 1727–1740, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. A. Pankov and G. Zhang, “Standing wave solutions for discrete nonlinear Schrödinger equations with unbounded potentials and saturable nonlinearity,” Journal of Mathematical Sciences, vol. 177, no. 1, pp. 71–82, 2011. View at Publisher · View at Google Scholar
  16. G. Zhang and A. Pankov, “Standing waves of the discrete nonlinear Schrödinger equations with growing potentials,” Communications in Mathematical Analysis, vol. 5, no. 2, pp. 38–49, 2008. View at Google Scholar · View at Zentralblatt MATH
  17. G. Zhang, “Breather solutions of the discrete nonlinear Schrödinger equations with unbounded potentials,” Journal of Mathematical Physics, vol. 50, no. 1, pp. 1–12, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. G. Zhang and F. Liu, “Existence of breather solutions of the DNLS equations with unbounded potentials,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 12, pp. e786–e792, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. Y. Ding and C. Lee, “Multiple solutions of Schrödinger equations with indefinite linear part and super or asymptotically linear terms,” Journal of Differential Equations, vol. 222, no. 1, pp. 137–163, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. Q. Li, H. Su, and Z. Wei, “Existence of infinitely many large solutions for the nonlinear Schrödinger-Maxwell equations,” Nonlinear Analysis. Theory, Methods & Applications, vol. 72, no. 11, pp. 4264–4270, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. D. S. Moschetto, “Existence and multiplicity results for a nonlinear stationary Schrödinger equation,” Annales Polonici Mathematici, vol. 99, no. 1, pp. 39–43, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. M.-H. Yang and Z.-Q. Han, “Existence and multiplicity results for the nonlinear Schrödinger-Poisson systems,” Nonlinear Analysis. Real World Applications, vol. 13, no. 3, pp. 1093–1101, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. M. Willem, Minimax Theorems, Birkhäuser, Boston, Mass, USA, 1996. View at Publisher · View at Google Scholar
  24. P. Rabinowitz, “Minimax methods in critical point theory with applications to differential equations,” vol. 65 of CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, USA, 1986. View at Google Scholar
  25. T. Bartsch, “Infinitely many solutions of a symmetric Dirichlet problem,” Nonlinear Analysis. Theory, Methods & Applications, vol. 20, no. 10, pp. 1205–1216, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  26. W. Zou, “Variant fountain theorems and their applications,” Manuscripta Mathematica, vol. 104, no. 3, pp. 343–358, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH