Skip to main content
SpringerLink
Log in

Existence of Weak Solutions for Generalized Quasilinear Schrödinger Equations

  • Published:
Journal of Dynamical and Control Systems Aims and scope Submit manuscript

Abstract

This paper shows the existence of nontrivial weak solutions for the generalized quasilinear Schrödinger equations

$$ -div(g^{p}(u)|\nabla u|^{p-2}\nabla u)+g^{p-1}(u)g^{\prime}(u)|\nabla u|^{p}+ V(x)|u|^{p-2}u=h(u),\,\, x\in \mathbb{R}^{N}, $$

where N ≥ 3, \(g(s): \mathbb {R}\rightarrow \mathbb {R}^{+}\) is C 1 nondecreasing function with respect to |s|, V is a positive potential bounded away from zero and h(u) is a nonlinear term of subcritical type. By introducing a variable replacement and using minimax methods, we show the existence of a nontrivial solution in \(C^{\alpha }_{loc}(\mathbb {R}^{N})\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bass F, Nasanov NN. Nonlinear electromagnetic spin waves. Phys Reports 1990; 189:165–223.

    Article  Google Scholar 

  2. Berestycki H, Gallouet T, Kavian O. Equations de Champs scalaires euclidiens non lineaires dans le plan. C R Acad Sci; Paris Ser I Math 1983;297(5):307–10.

    MathSciNet  MATH  Google Scholar 

  3. Berestycki H, Gallouet T, Kavian O. 1984. Publications du Laboratoire d’Analyse Num6rique, Universit6 de Paris VI, MR 85e:35041.

  4. Berestycki H, Lions PL. Nonlinear scalar field equations I: existence of a state. Arch Rat Mech Anal 1983;82:313–46. MR 84h:35054a.

    MathSciNet  MATH  Google Scholar 

  5. Chen XL, Sudan RN. Necessary and sufficient conditions for self-focusing of short ultraintense laser pulse. Phys Rev Lett 1993;70:2082–5.

    Article  Google Scholar 

  6. Colin M. Stability of stationary waves for a quasilinear Schrödinger equation in space dimension 2. Adv in Diff Eqs 2003;8(1):1–28.

    MathSciNet  MATH  Google Scholar 

  7. Colin M, Jeanjean L. Solutions for a quasilinear Schrödinger equation: a dual approach. Nonlinear Anal 2004;56:213–226.

    Article  MathSciNet  MATH  Google Scholar 

  8. do Ó JM, de Medeiros ES. Remarks on least energy solutions for quasilinear elliptic problems in \(\mathbb {R}^{N}\). Electron J Differential Equations 2003;83:14.

    Google Scholar 

  9. do Ó JM, Miyagaki O, Soares S. Soliton solutions for quasilinear Schrödinger equations: the critical exponential case. Nonlinear Anal 2007;67:3357–3372.

    Article  MathSciNet  MATH  Google Scholar 

  10. Evans LC. Partial differential equations, graduate studies in Mathematics. Amer Math Soc. 1998;19:261–76.

  11. Fleming WH. A selection-migration model in population genetic. J Math Bio 1975; 20:219–233.

    Article  MathSciNet  Google Scholar 

  12. Hasse RW. A general method for the solution of nonlinear soliton and kink Schrödinger equation. Z Phys B 1980;37:83–7.

    Article  MathSciNet  Google Scholar 

  13. Kelley PL. Self focusing of optical beams. Phys Rev Lett 1965;15:1005–8.

    Article  Google Scholar 

  14. Kurihura S. Large-amplitude quasi-solitons in superfluids films. J Phys Soc Japan 1981;50:3262–7.

    Article  Google Scholar 

  15. Laedke EW, Spatschek KH, Stenflo L. Evolution theorem for a class of perturbed envelope soliton solutions. J Math Phys 1983;24:2764–9.

    Article  MathSciNet  MATH  Google Scholar 

  16. Lions PL. The concentration compactness principle in the calculus of variations, the locally compact case, parts I and II. Ann Inst H Poincaré Anal Non Lineairé 1984; 109–145:223–83.

    Google Scholar 

  17. Liu J, Wang Y, Wang ZQ. Solutions for a quasilinear Schrödinger equation via the Nehari Method. Comm Partial Differential Equations 2004;29:879–901.

    Article  MathSciNet  MATH  Google Scholar 

  18. Makhankov VG, Fedyanin VK. Non-linear effects in quasi-one-dimensional models of condensed matter theory. Phys Reports 1984;104:1–86.

    Article  MathSciNet  Google Scholar 

  19. Ritchie B. Relativistic self-focusing and channel formation in laser-plasma interactions. Phys Rev E 1994;50:687–9.

    Article  Google Scholar 

  20. Schechter M. Linking Methods in Critical Point Theory. Boston: Birkhäuser; 1999.

    Book  MATH  Google Scholar 

  21. Serrin J. Local behavior of solutions of quasi-linear equations. Acta Math 1964; 111:247–302.

    Article  MathSciNet  MATH  Google Scholar 

  22. Takeno S, Homma S. Classical planar Heinsenberg ferromagnet, complex scalar fields and nonlinear excitations. Progr Theoret Physics 1981;65:172–89.

    Article  MathSciNet  Google Scholar 

  23. Tolksdorff P. Regularity for a more general class of quasilinear elliptic equations. J Differential Equations 1984;51:126–50.

    Article  MathSciNet  Google Scholar 

  24. Trudinger NS. On Harnack type inequalities and their applications to quasilinear elliptic equations. Comm Pure Appl Math 1967;20:721–47.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors would like to express their sincere gratitude to the anonymous reviewer for the valuable comments and suggestions.

Author information

Authors and Affiliations

  1. College of Science, Hohai University, Nanjing, 210098, People’s Republic of China

    Hongxue Song & Caisheng Chen

  2. College of Science, Nanjing University of Posts and Telecommunications, Nanjing, 210023, People’s Republic of China

    Hongxue Song

Authors
  1. Hongxue Song
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Caisheng Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hongxue Song.

Additional information

This work is partly supported by the National Natural Science Foundations of China (Grant No. 11401165) and the National Natural Science Foundations of China (Grant No. 11201241).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Song, H., Chen, C. Existence of Weak Solutions for Generalized Quasilinear Schrödinger Equations. J Dyn Control Syst 22, 369–383 (2016). https://doi.org/10.1007/s10883-015-9298-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10883-015-9298-z

Keywords

Keywords

Search

Navigation