Journal of Applied Mathematics

Existence of Solutions for a Modified Nonlinear Schrödinger System

Yujuan Jiao and Yanli Wang

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We are concerned with the following modified nonlinear Schrödinger system: -Δu+u-(1/2)uΔ(u2)=(2α/(α+β))|u|α-2|v|βu,  xΩ,  -Δv+v-(1/2)vΔ(v2)=(2β/(α+β))|u|α|v|β-2v,  xΩ,  u=0,  v=0,  xΩ, where α>2,  β>2,  α+β<2·2*,  2*=2N/(N-2) is the critical Sobolev exponent, and ΩN  (N3) is a bounded smooth domain. By using the perturbation method, we establish the existence of both positive and negative solutions for this system.

Article information

Source
J. Appl. Math. Volume 2013 (2013), Article ID 431672, 9 pages.

Dates
First available in Project Euclid: 14 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.jam/1394808044

Digital Object Identifier
doi:10.1155/2013/431672

Mathematical Reviews number (MathSciNet)
MR3074319

Zentralblatt MATH identifier
1271.35065

Citation

Jiao, Yujuan; Wang, Yanli. Existence of Solutions for a Modified Nonlinear Schrödinger System. J. Appl. Math. 2013 (2013), Article ID 431672, 9 pages. doi:10.1155/2013/431672. https://projecteuclid.org/euclid.jam/1394808044


Export citation

References

  • S. Kurihara, “Large-amplitude quasi-solitons in superfluid films,” Journal of the Physical Society of Japan, vol. 50, no. 10, pp. 3262–3267, 1981.
  • E. W. Laedke, K. H. Spatschek, and L. Stenflo, “Evolution theorem for a class of perturbed envelope soliton solutions,” Journal of Mathematical Physics, vol. 24, no. 12, pp. 2764–2769, 1983.
  • H. S. Brandi, C. Manus, G. Mainfray, T. Lehner, and G. Bonnaud, “Relativistic and ponderomotive self-focusing of a laser beam in a radially inhomogeneous plasma. I: paraxial approximation,” Physics of Fluids B, vol. 5, no. 10, pp. 3539–3550, 1993.
  • X. L. Chen and R. N. Sudan, “Necessary and sufficient conditions for self-focusing of short ultraintense laser pulse in underdense plasma,” Physical Review Letters, vol. 70, no. 14, pp. 2082–2085, 1993.
  • A. de Bouard, N. Hayashi, and J. C. Saut, “Global existence of small solutions to a relativistic nonlinear Schrödinger equation,” Communications in Mathematical Physics, vol. 189, no. 1, pp. 73–105, 1997.
  • B. Ritchie, “Relativistic self-focusing and channel formation in laser-plasma interactions,” Physical Review E, vol. 50, no. 2, pp. R687–R689, 1994.
  • M. Poppenberg, K. Schmitt, and Z. Q. Wang, “On the existence of soliton solutions to quasilinear Schrödinger equations,” Calculus of Variations and Partial Differential Equations, vol. 14, no. 3, pp. 329–344, 2002.
  • J. Q. Liu, Y. Q. Wang, and Z. Q. Wang, “Soliton solutions for quasilinear Schrödinger equations II,” Journal of Differential Equations, vol. 187, no. 2, pp. 473–493, 2003.
  • M. Colin and L. Jeanjean, “Solutions for a quasilinear Schrödinger equation: a dual approach,” Nonlinear Analysis A, vol. 56, no. 2, pp. 213–226, 2004.
  • H. Berestycki and P. L. 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.
  • 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.
  • Y. Guo and Z. Tang, “Ground state solutions for quasilinear Schrödinger systems,” Journal of Mathematical Analysis and Applications, vol. 389, no. 1, pp. 322–339, 2012.
  • P. L. Lions, “The concentration-compactness principle in the calculus of variations. The locally compact case I,” Annales de l'Institut Henri Poincaré, vol. 1, no. 2, pp. 109–145, 1984.
  • Y. Guo and Z. Tang, “Ground state solutions for the quasilinear Schrödinger equation,” Nonlinear Analysis A, vol. 75, no. 6, pp. 3235–3248, 2012.
  • T. Bartsch and Z. Q. Wang, “Multiple positive solutions for a nonlinear Schrödinger equation,” Zeitschrift für Angewandte Mathematik und Physik, vol. 51, no. 3, pp. 366–384, 2000.
  • A. Ambrosetti, M. Badiale, and S. Cingolani, “Semiclassical states of nonlinear Schrödinger equations,” Archive for Rational Mechanics and Analysis, vol. 140, no. 3, pp. 285–300, 1997.
  • A. Ambrosetti, A. Malchiodi, and S. Secchi, “Multiplicity results for some nonlinear Schrödinger equations with potentials,” Archive for Rational Mechanics and Analysis, vol. 159, no. 3, pp. 253–271, 2001.
  • J. Byeon and Z. Q. Wang, “Standing waves with a critical frequency for nonlinear Schrödinger equations II,” Calculus of Variations and Partial Differential Equations, vol. 18, no. 2, pp. 207–219, 2003.
  • S. Cingolani and M. Lazzo, “Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions,” Journal of Differential Equations, vol. 160, no. 1, pp. 118–138, 2000.
  • S. Cingolani and M. Nolasco, “Multi-peak periodic semiclassical states for a class of nonlinear Schrödinger equations,” Proceedings of the Royal Society of Edinburgh A, vol. 128, no. 6, pp. 1249–1260, 1998.
  • M. Del Pino and P. L. Felmer, “Multi-peak bound states for nonlinear Schrödinger equations,” Annales de l'Institut Henri Poincaré, vol. 15, no. 2, pp. 127–149, 1998.
  • M. del Pino and P. L. Felmer, “Semi-classical states for nonlinear Schrödinger equations,” Journal of Functional Analysis, vol. 149, no. 1, pp. 245–265, 1997.
  • A. Floer and A. Weinstein, “Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential,” Journal of Functional Analysis, vol. 69, no. 3, pp. 397–408, 1986.
  • Y. G. Oh, “On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential,” Communications in Mathematical Physics, vol. 131, no. 2, pp. 223–253, 1990.
  • Y. G. Oh, “Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class ${(V)}_{a}$,” Communications in Partial Differential Equations, vol. 13, no. 12, pp. 1499–1519, 1988.
  • X. Q. Liu, J. Q. Liu, and Z. Q. Wang, “Quasilinear elliptic equations via perturbation method,” Proceedings of the American Mathematical Society, vol. 141, no. 1, pp. 253–263, 2013.
  • J. Liu and Z. Q. Wang, “Soliton solutions for quasilinear Schrödinger equations I,” Proceedings of the American Mathematical Society, vol. 131, no. 2, pp. 441–448, 2003.
  • A. Canino and M. Degiovanni, “Nonsmooth critical point theory and quasilinear elliptic equations,” in Topological Methods in Differential Equations and Inclusions, P. Q. Montreal, Ed., vol. 472 of Nato Advanced Study Institute Series C: Mathematical and Physical Sciences, pp. 1–50, Kluwer Academic, 1995.
  • J. Q. Liu and Z. Q. Wang, “Bifurcations for quasilinear elliptic equations II,” Communications in Contemporary Mathematics, vol. 10, no. 5, pp. 721–743, 2008.