Journal of Dynamical and Control Systems

, Volume 25, Issue 1, pp 79–94 | Cite as

Existence and Multiplicity of Standing Wave Solutions for a Class of Quasilinear Schrödinger Systems in \(\mathbb {R}^{N}\)

  • Hongxue SongEmail author
  • Caisheng Chen
  • Wei Liu


In this paper, we study the following quasilinear Schrödinger systems of the form
$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{l} -\triangle_{p} u_{j}+a_{j}(x)|u_{j}|^{p-2}u_{j}\,-\,\triangle_{p} (|u_{j}|^{2})u_{j}\,=\,\mu_{j}|u_{j}|^{q-2}u_{j}\,+\,\frac{1}{2}{\sum}_{i\neq j}\beta_{ij}|u_{i}|^{m}|u_{j}|^{m-2}u_{j}, ~x\!\in\! \mathbb{R}^{N},\\ u_{j}(x)\rightarrow 0,~\text{as} ~|x|\rightarrow \infty, \ j = 1,\ \cdots,\ k, \end{array} \right. \end{array} $$
where N ≥ 3, 2 ≤ pN, and the potential aj(x) is positive and bounded in \(\mathbb {R}^{N}\), μj > 0, βij = βji for 1 ≤ i < jk(k ≥ 2). Using symmetric mountain pass lemma, we obtain infinitely many solutions to Schrödinger system (0.1).


Quasilinear Schrödinger systems Dual approach Mountain pass lemma 

Mathematics Subject Classification (2010)

35J50 35J70 35J92 


  1. 1.
    Aires JFL, Souto MAS. Existence of solutions for a quasilinear schrödinger equation with potential vanishing. J Math Anal Appl 2014;416:924–946.MathSciNetCrossRefGoogle Scholar
  2. 2.
    Alves CO, Souto MAS. Existence of solutions for a class of nonlinear schrödinger equations with potential vanishing at infinity. J Differ Equations 2013;254:1977–1991.CrossRefGoogle Scholar
  3. 3.
    Ambrosetti A, Colorado E. Bound and ground states of coupled nonlinear schrödinger equations. C R Math Acad Sci Paris 2006;342:453–458.MathSciNetCrossRefGoogle Scholar
  4. 4.
    Ambrosetti A, Colorado E. Standing waves of some coupled nonlinear schrödinger equations. J Lond Math Soc 2007;75:67–82.MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bartsch T, Dancer EN, Wang ZQ. A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system. Calc Var PDEs 2010;37:345–361.MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bartsch T, Wang ZQ. Note on ground states of nonlinear schrödinger systems. J Partial Differ Equ 2006;19:200–207.zbMATHGoogle Scholar
  7. 7.
    Bartsch T, Wang ZQ, Wei J. Bound states for a coupled schrödinger system. J Fixed Point Theory Appl 2007;2:353–367.MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bass F, Nasanov NN. Nonlinear electromagnetic spin waves. Phys Rep 1990; 189:165–223.CrossRefGoogle Scholar
  9. 9.
    Brezis H, Lieb EH. A relation between pointwise convergence of functions and convergence of functionals. Proc Amer Math Soc 1983;88:486–490.MathSciNetCrossRefGoogle Scholar
  10. 10.
    Caffarelli L, Kohn R, Nirenberg L. First order interpolation inequalities with weights. Compos Math 1984;53:259–275.MathSciNetzbMATHGoogle Scholar
  11. 11.
    Chang S, Lin CS, Lin TC, Lin W. Segregated nodal domains of two-dimensional multispecies Bose-Einstein condensates. Phys D 2004;196:341–361.MathSciNetCrossRefGoogle Scholar
  12. 12.
    Chen CS. Multiple solutions for a class of quasilinear Schrödinger equations in \(\mathbb {R}^{N}\). J Math Phys 2015;56:071507. Scholar
  13. 13.
    Chen SX. Existence of positive solutions for a class of quasilinear schrödinger equations on \(\mathbb {R}^{N}\). J Math Anal Appl 2013;405:595–607.MathSciNetCrossRefGoogle Scholar
  14. 14.
    Colin M, Jeanjean L. Solutions for a quasilinear schrödinger equation: a dual approach. Nonlinear Anal 2004;56:213–226.MathSciNetCrossRefGoogle Scholar
  15. 15.
    Dancer EN, Wei J, Weth T. A priori bounds versus multiple existence of positive solutions for a nonlinear schrödinger system. Ann I H Poincaré 2010;27: 953–969.CrossRefGoogle Scholar
  16. 16.
    Evans LC. Partial differential equations, graduate studies in mathematics. Amer Math Soc 1998;19:261–276.Google Scholar
  17. 17.
    Guo Y, Tang Z. Ground state solutions for quasilinear schrödinger systems. J Math Anal Appl 2012;389:322–339.MathSciNetCrossRefGoogle Scholar
  18. 18.
    Hasse RW. A general method for the solution of nonlinear soliton and kink schrödinger equation. Z Phys B 1980;37:83–87.MathSciNetCrossRefGoogle Scholar
  19. 19.
    Kurihura S. Large-amplitude quasi-solitons in superfluids films. J Phys Soc Japan 1981;50:3262–3267.CrossRefGoogle Scholar
  20. 20.
    Lin TC, Wei JC. Ground state of N Coupled Nonlinear Schrö,dinger equations in \(\mathbb {R}^{n}\), n3. Commun Math Phys 2005;255:629–653.CrossRefGoogle Scholar
  21. 21.
    Liu JQ, Liu XQ, Wang ZQ. Multiple mixed states of nodal solutions for nonlinear schrödinger systems. Calc Var 2015;52:565–586.MathSciNetCrossRefGoogle Scholar
  22. 22.
    Liu JQ, Wang Y, Wang ZQ. Solutions for quasilinear schrödinger equations, II. J Differential Equations 2003;187:473–493.MathSciNetCrossRefGoogle Scholar
  23. 23.
    Liu JQ, Wang Y, Wang ZQ. Solutions for a quasilinear schrödinger equation via the Nehari Method. Comm Partial Differential Equations 2004;29:879–901.MathSciNetCrossRefGoogle Scholar
  24. 24.
    Liu JQ, Wang ZQ. Soliton solutions for quasilinear schrödinger equations. Proc Amer Math Soc 2003;131:441–448.MathSciNetCrossRefGoogle Scholar
  25. 25.
    Liu Z, Wang ZQ. Multiple bound states of nonlinear schrödinger systems. Comm Math Phys 2008;282:721–731.MathSciNetCrossRefGoogle Scholar
  26. 26.
    Maia LA, Montefusco E, Pellacci B. Positive solutions for a weakly coupled nonlinear schrödinger system. J Diff Equ 2006;299:743–767.CrossRefGoogle Scholar
  27. 27.
    Mitchell M, Segev M. Self-trapping of inconherentwhite light. Nature 1997; 387:880–882.CrossRefGoogle Scholar
  28. 28.
    Montefusco E, Pellacci B, Squassina M. Semiclassical states for weakly coupled nonlinear schrödinger systems. J European Math Soc 2008;10:41–71.zbMATHGoogle Scholar
  29. 29.
    Noris B, Tavares H, Terracini S, Verzini G. Uniform hölder bounds for nonlinear schrödinger systems with strong competition. Comm Pure and Appl Math 2010;63:267–302.MathSciNetCrossRefGoogle Scholar
  30. 30.
    Pomponio A. Coupled nonlinear Schrödinger systems with potentials. J Differential Equations 2006;227:258–281.MathSciNetCrossRefGoogle Scholar
  31. 31.
    Rabinowitz PH, Vol. 65. In minimax methods in critical point theory with application to differential equations CBMS regional conference series in mathematics. Providence, RI: American Mathematical Society; 1986.CrossRefGoogle Scholar
  32. 32.
    Ritchie B. Relativistic self-focusing and channel formation in laser-plasma interactions. Phys Rev E 1994;50:687–689.CrossRefGoogle Scholar
  33. 33.
    Rüegg Ch, et al. Bose-Einstein condensation of the triple states in the magnetic insulator TlCuCl3. Nature 2003;423:62–65.CrossRefGoogle Scholar
  34. 34.
    Sato Y, Wang ZQ. On the least energy sign-changing solutions for a nonlinear elliptic system. Discrete Cont Dyn 2015;35:2151–2164.MathSciNetCrossRefGoogle Scholar
  35. 35.
    Severo U. Existence of weak solutions for quasilinear elliptic equations involving the p-Laplacian. EJDE 2008;2008:1–16.MathSciNetzbMATHGoogle Scholar
  36. 36.
    Sirakov B. Least energy solitary waves for a system of nonlinear schrödinger equations in \(\mathbb {R}^{n}\). Comm Math Phys 2007;271:199–221.MathSciNetCrossRefGoogle Scholar
  37. 37.
    Stavrakakis NM, Zographopoulos NB. Multiplicity and regularity results for some quasilinear elliptic systems on \(\mathbb {R}^{N}\). Nonlinear Anal 2002;50:55–69.MathSciNetCrossRefGoogle Scholar
  38. 38.
    Terracini S, Verzini G. Multipulse phase in k-mixtures of Bose-Einstein condensates. Arch Rat Mech Anal 2009;194:717–741.MathSciNetCrossRefGoogle Scholar
  39. 39.
    Tian R, Wang ZQ. Multiple solitary wave solutions of nonlinear schrödinger systems. Topo Meth Non Anal 2011;37:203–223.zbMATHGoogle Scholar
  40. 40.
    Willem M, Vol. 24. Minimax theorems, progr. Nonlinear differential equations Appl. Boston: Birkhäuser Boston, Inc; 1996.Google Scholar
  41. 41.
    Wu X. Multiple solutions for quasilinear schrödinger equations with a parameter. J Differential Equations 2014;256:2619–2632.MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.College of ScienceHohai UniversityNanjingPeople’s Republic of China
  2. 2.College of ScienceNanjing University of Posts and TelecommunicationsNanjingPeople’s Republic of China

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