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Abstract and Applied Analysis
Volume 2014, Article ID 317571, 8 pages
http://dx.doi.org/10.1155/2014/317571
Research Article

Ground States for the Schrödinger Systems with Harmonic Potential and Combined Power-Type Nonlinearities

School of Mathematics and Physics, University of Science and Technology Beijing, 30 Xueyuan Road, Haidian District, Beijing 100083, China

Received 14 June 2014; Accepted 28 August 2014; Published 19 October 2014

Academic Editor: Vladimir Georgiev

Copyright © 2014 Baiyu Liu. 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. L. P. Pitaevskii and S. Stringari, Bose-Einstein Condensation, Clarendon Press, Oxford, UK, 2003. View at MathSciNet
  2. T. Bartsch and Z. Q. Wang, “Existence and multiple results for some superlinear elliptic problems on RN,” Communications in Partial Differential Equations, vol. 20, pp. 1725–1741, 1995. View at Google Scholar
  3. D. G. Costa, “On a class of elliptic systems in RN,” Electronic Journal of Differential Equations, vol. 7, pp. 1–14, 1994. View at Google Scholar · View at MathSciNet
  4. A. Ambrosetti and E. Colorado, “Standing waves of some coupled nonlinear Schröinger equations,” Journal of the London Mathematical Society, vol. 75, no. 1, pp. 67–82, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. H. Berestycki and P. Lions, “Nonlinear scalar field equations. I. Existence of a ground state,” Archive for Rational Mechanics and Analysis, vol. 82, no. 4, pp. 313–345, 1983. View at Publisher · View at Google Scholar · View at MathSciNet
  6. W. Y. Ding and W. M. Ni, “On the existence of positive entire solutions of a semilinear elliptic equation,” Archive for Rational Mechanics and Analysis, vol. 91, no. 4, pp. 283–308, 1986. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. V. Georgiev and G. Venkov, “Symmetry and uniqueness of minimizers of Hartree type equations with external Coulomb potential,” Journal of Differential Equations, vol. 251, no. 2, pp. 420–438, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  8. B. Gidas, W. M. Ni, and L. Nirenberg, “Symmetry of positive solutions of nonlinear elliptic equations in RN,” Journal of Mathematical Analysis and Applications A, vol. 7, pp. 369–402, 1981. View at Google Scholar
  9. N. Ikoma and K. Tanaka, “A local mountain pass type result for a system of nonlinear Schrödinger equations,” Calculus of Variations and Partial Differential Equations, vol. 40, no. 3-4, pp. 449–480, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. T. C. Lin and J. C. Wei, “Ground state of N coupled nonliear Schrödinger equations in Rn,n3,” Communications in Mathematical Physics, vol. 255, pp. 629–653, 2005. View at Google Scholar
  11. T. C. Lin and J. C. Wei, “Spikes in two-component systems of nonlinear Schröinger equations with trapping potentials,” Journal of Differential Equations, vol. 229, no. 2, pp. 538–569, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. L. Ma and L. Zhao, “Uniqueness of ground states of some coupled nonlinear Schrödinger systems and their application,” Journal of Differential Equations, vol. 245, no. 9, pp. 2551–2565, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. L. A. Maia, E. Montefusco, and B. Pellacci, “Positive solutions for a weakly coupled nonlinear Schrödinger system,” Journal of Differential Equations, vol. 229, no. 2, pp. 743–767, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. 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
  15. G. M. Wei, “Existence and concentration of ground states of coupled nonlinear Schrödinger equations,” Journal of Mathematical Analysis and Applications, vol. 332, no. 2, pp. 846–862, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  16. M. Lucia and Z. Tang, “Multi-bump bound states for a system of nonlinear Schrödinger equations,” Journal of Differential Equations, vol. 252, no. 5, pp. 3630–3657, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. B. Sirakov, “Least energy solitary waves for a system of nonlinear schröinger equations in Rn,” Communications in Mathematical Physics, vol. 271, no. 1, pp. 199–221, 2007. View at Publisher · View at Google Scholar
  18. X. Song, “Stability and instability of standing waves to a system of Schrödinger equations with combined power-type nonlinearities,” Journal of Mathematical Analysis and Applications, vol. 366, no. 1, pp. 345–359, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  19. W. A. Strauss, “Existence of solitary waves in higher dimensions,” Communications in Mathematical Physics, vol. 55, no. 2, pp. 149–162, 1977. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  20. R. Fukuizumi and T. Ozawa, “Exponential decay of solutions to nonlinear elliptic equations with potentials,” Zeitschrift für angewandte Mathematik und Physik ZAMP, vol. 56, no. 6, pp. 1000–1011, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. T. Cazenave, An Introduction to Nonlinear Schrödinger Equations, 1993.
  22. N. Okazawa, “An Lp theory for Schrödinger operators with nonnegative potentials,” Journal of the Mathematical Society of Japan, vol. 36, no. 4, pp. 675–688, 1984. View at Publisher · View at Google Scholar · View at MathSciNet
  23. B. Y. Liu and L. Ma, “Symmetry and uniqueness result for the Schrödinger system with potential,” preprint.
  24. J. Busca and B. Sirakov, “Symmetry results for semilinear elliptic systems in the whole space,” Journal of Differential Equations, vol. 163, no. 1, pp. 41–56, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus