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Abstract and Applied Analysis
Volume 2014, Article ID 607078, 7 pages
http://dx.doi.org/10.1155/2014/607078
Research Article

Nehari-Type Ground State Positive Solutions for Superlinear Asymptotically Periodic Schrödinger Equations

1Department of Mathematics, Huaihua College, Huaihua, Hunan 418008, China
2School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China

Received 3 January 2014; Revised 23 May 2014; Accepted 11 June 2014; Published 29 June 2014

Academic Editor: Marco Squassina

Copyright © 2014 Xiaoyan Lin and X. H. Tang. 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. S. Alama and Y. Y. Li, “On “multibump” bound states for certain semilinear elliptic equations,” Indiana University Mathematics Journal, vol. 41, no. 4, pp. 983–1026, 1992. View at Publisher · View at Google Scholar · View at MathSciNet
  2. S. Angenent, “The shadowing lemma for elliptic PDE,” in Dynamics of Infinite Dimensional Systems, vol. 37 of NATO ASI Series, pp. 7–22, Springer, Berlin, Germany, 1986. View at Publisher · View at Google Scholar
  3. T. Bartsch and Y. Ding, “Homoclinic solutions of an infinite-dimensional Hamiltonian system,” Mathematische Zeitschrift, vol. 240, no. 2, pp. 289–310, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. T. Bartsch, A. Pankov, and Z. Wang, “Nonlinear Schrödinger equations with steep potential well,” Communications in Contemporary Mathematics, vol. 3, no. 4, pp. 549–569, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  5. J. Chabrowski and A. Szulkin, “On a semilinear Schrödinger equation with critical Sobolev exponent,” Proceedings of the American Mathematical Society, vol. 130, no. 1, pp. 85–93, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. Y. Deng, L. Jin, and S. Peng, “Solutions of Schrödinger equations with inverse square potential and critical nonlinearity,” Journal of Differential Equations, vol. 253, no. 5, pp. 1376–1398, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. Y. Ding and A. Szulkin, “Bound states for semilinear Schrödinger equations with sign-changing potential,” Calculus of Variations and Partial Differential Equations, vol. 29, no. 3, pp. 397–419, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. Y. Ding and S. X. Luan, “Multiple solutions for a class of nonlinear Schrödinger equations,” Journal of Differential Equations, vol. 207, no. 2, pp. 423–457, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. 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 MathSciNet · View at Scopus
  10. W. Kryszewski and A. Szulkin, “Generalized linking theorem with an application to a semilinear Schrödinger equation,” Advances in Differential Equations, vol. 3, no. 3, pp. 441–472, 1998. View at Google Scholar · View at MathSciNet
  11. W. Kryszewski and A. Szulkin, “Infinite-dimensional homology and multibump solutions,” Journal of Fixed Point Theory and Applications, vol. 5, no. 1, pp. 1–35, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  12. Y. Q. Li, Z. Q. Wang, and J. Zeng, “Ground states of nonlinear Schrödinger equations with potentials,” Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire, vol. 23, no. 6, pp. 829–837, 2006. View at Google Scholar
  13. J. Q. Liu, Y. Q. Wang, and Z. Q. Wang, “Solutions for quasilinear Schrödinger equations via the Nehari method,” Communications in Partial Differential Equations, vol. 29, no. 5-6, pp. 879–901, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. Z. Liu and Z.-Q. Wang, “On the Ambrosetti-Rabinowitz superlinear condition,” Advanced Nonlinear Studies, vol. 4, pp. 561–572, 2004. View at Google Scholar
  15. A. Pankov, “Periodic nonlinear Schrödinger equation with application to photonic crystals,” Milan Journal of Mathematics, vol. 73, pp. 259–287, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, vol. 65 of Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, USA, 1986.
  17. P. H. Rabinowitz, “On a class of nonlinear Schrödinger equations,” Zeitschrift für Angewandte Mathematik und Physik, vol. 43, no. 2, pp. 270–291, 1992. View at Publisher · View at Google Scholar · View at MathSciNet
  18. M. Schechter, Minimax Systems and Critical Point Theory, Birkhäuser, Boston, Mass, USA, 2009. View at MathSciNet
  19. M. Schechter and W. Zou, “Double linking theorem and multiple solutions,” Journal of Functional Analysis, vol. 205, no. 1, pp. 37–61, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  20. A. Szulkin and T. Weth, “Ground state solutions for some indefinite variational problems,” Journal of Functional Analysis, vol. 257, no. 12, pp. 3802–3822, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. X. H. Tang, “Infinitely many solutions for semilinear Schrödinger equations with sign-changing potential and nonlinearity,” Journal of Mathematical Analysis and Applications, vol. 401, no. 1, pp. 407–415, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. X. H. Tang, “New super-quadratic conditions on ground state solutions for superlinear Schrödinger equation,” Advanced Nonlinear Studies, vol. 14, pp. 349–361, 2014. View at Google Scholar · View at MathSciNet
  23. X. H. Tang, “New conditions on nonlinearity for a periodic Schrödinger equation having zero as spectrum,” Journal of Mathematical Analysis and Applications, vol. 413, no. 1, pp. 392–410, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  24. C. Troestler and M. Willem, “Nontrivial solution of a semilinear Schrödinger equation,” Communications in Partial Differential Equations, vol. 21, no. 9-10, pp. 1431–1449, 1996. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. F. A. van Heerden, “Homoclinic solutions for a semilinear elliptic equation with an asymptotically linear nonlinearity,” Calculus of Variations and Partial Differential Equations, vol. 20, no. 4, pp. 431–455, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  26. M. Willem, Minimax Theorems, Birkhäuser, Boston, Mass, USA, 1996. View at Publisher · View at Google Scholar · View at MathSciNet
  27. M. Yang, “Ground state solutions for a periodic Schrödinger equation with superlinear nonlinearities,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 5, pp. 2620–2627, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  28. M. Yang, W. Chen, and Y. Ding, “Solutions for periodic Schrödinger equation with spectrum zero and general superlinear nonlinearities,” Journal of Mathematical Analysis and Applications, vol. 364, no. 2, pp. 404–413, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  29. P. L. Lions, “The concentration-compactness principle in the calculus of variations. The locally compact case. I,” Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, vol. 1, pp. 109–145, 1984. View at Google Scholar
  30. M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer, Berlin, Germany, 2000. View at Publisher · View at Google Scholar · View at MathSciNet