Skip to main content
SpringerLink
Log in

The existence of semiclassical states for some p-Laplacian equation with critical exponent

  • Published:
Acta Mathematicae Applicatae Sinica, English Series Aims and scope Submit manuscript

Abstract

In this paper, we study the existence of semiclassical states for some p-Laplacian equation. Under given conditions and minimax methods, we show that this problem has at least one positive solution provided that ε ≤ E; for any m ∈ ℕ, it has m pairs solutions if ε ≤ E m , where E, Em are sufficiently small positive numbers. Moreover, these solutions are closed to zero in W1,p(ℝN) as ε → 0.

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. Ambrosetti, A., Badiale, M., Cingolani, S. Semiclassical States of Nonlinear Schrödinger Equations. Arch. Ration. Mech. Anal., 140: 285–300 (1997)

    Article  MATH  Google Scholar 

  2. Alves, C.O., Carriao, P.C., Medeiros, E.S. Multiplicity of solutions for a class of quasilinear problem in exterior domains with Neumann conditions. Abstr. Appl. Anal., 3: 251–268 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alves, C.O., Figueiredo, G.M. in RN.Differ. Integral Equ., 19: 143–162 (2006)

    MATH  Google Scholar 

  4. Brandi, H., Manus, C., Mainfray, G., Lehner, T., Bonnaud, G. Relativistic and ponderomotive self-focusing of a laser beam in a radially inhomogeneous plasma. Phys. Fluids B, 5: 3539–3550 (1993)

    Article  Google Scholar 

  5. Benci, V. On critical point theory for indefinite functionals in the presence of symmetries. Trans. Amer. Math. Soc., 274: 533–572 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bass, F.G., Nasanov, N.N. Nonlinear electromagnetic-spin waves. Phys. Rep., 189: 165–223 (1990)

    Article  Google Scholar 

  7. Brüll, L., Lange, H. Solitary waves for quasilinear Schrödinger equations. Exposition. Math., 4: 279–288 (1986)

    MathSciNet  MATH  Google Scholar 

  8. Chen, X.L., Sudan, R.N. Necessary and sufficient conditions for self-focusing of short ultraintense laser pulse in underdense plasma. Phys. Rev. Lett., 70: 2082–2085 (1993)

    Article  Google Scholar 

  9. Colin, M., Jeanjean, L. Solutions for a quasilinear Schrödinger equation: a dual approach. Nonlinear Anal. TMA., 56: 213–226 (2004)

    Article  MATH  Google Scholar 

  10. do Ó, J.M., Medeiros, E., Severo, U. On a quasilinear nonhomogeneous elliptic equation with critical growth in RN.J. Differ. Equ., 246: 1363–1386 (2009)

    Article  MATH  Google Scholar 

  11. do Ó, J.M., Miyagakib, O., Soares, S. Soliton solutions for quasilinear Schrödinger equations with critical growth. J. Differ. Equ., 248: 722–744 (2010)

    Article  MATH  Google Scholar 

  12. de Bouard, A., Hayashi, N., Saut., J. Global existence of small solutions to a relativistic nonlinear Schrödinger equation. Comm. Math. Phys., 189: 73–105 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  13. Grossi, M. Some results on a class of nonlinear Schrödinger equations. Math. Z., 235: 687–705 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kosevich, A.M., Ivanov, B.A, Kovalev, A.S. Magnetic solitons. Phys. Rep., 194: 117–238 (1990)

    Article  Google Scholar 

  15. Kurihara, S. Large-amplitude quasi-solitons in superfluid films. J.Phys. Soc. Japan, 50: 3262–3267 (1981)

    Article  Google Scholar 

  16. Kang, X., Wei, J. On interacting bumps of semi-classical states of nonlinear Schrödinger equations. Adv. Differ. Equ., 5: 899–928 (2000)

    MATH  Google Scholar 

  17. Liu, J., Wang, Z. Soliton solutions for quasilinear Schrödinger equations, I. Proc. Amer. Math. Soc., 131: 441–448 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  18. Liu, J., Wang, Z. Symmetric solutions to a modified nonlinear Schrödinger equation. Nonlinearity, 21: 121–133 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Liu, J., Wang, Y., Wang, Z. Soliton solutions for quasilinear Schrödinger equations, II.J. Differ. Equ., 187: 473–493 (2003)

    Article  MATH  Google Scholar 

  20. Liu, J., Wang, Y., Wang, Z. Solutions for quasilinear Schrödinger equations via the Nehari method. Comm. Partial Differ. Equ., 29: 879–901 (2004)

    Article  MATH  Google Scholar 

  21. Li, Y. On a singularly perturbed elliptic equation. Adv. Differ. Equ., 2: 955–980 (1997)

    MathSciNet  MATH  Google Scholar 

  22. Moameni, A. Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in RN. J. Differ. Equ., 229: 570–587 (2006)

    Article  MATH  Google Scholar 

  23. del Pino, M., Felmer, P. Semi-classical states of nonlinear Schrödinger equations: a variational reduction method. Math. Ann., 324: 1–32 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  24. Poppenberg, M., Schmitt, K., Wang, Z. On the existence of soliton solutions to quasilinear Schrödinger equations. Calc. Var. Partial Differ. Equ., 14: 329–344 (2002)

    Article  MATH  Google Scholar 

  25. Quispel, G.R.W., Capel, H.W. Equation of motion for the Heisenberg spin chain. Phys. A, 110: 41–80 (1982)

    Article  MathSciNet  Google Scholar 

  26. Ritchie, B. Relativistic self-focusing and channel formation in laser-plasma interactions. Phys. Rev. E, 50: 687–689 (1994)

    Article  Google Scholar 

  27. Rabinowitz, P.H. On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys., 43: 270–291 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  28. Severo, U. Existence of weak solutions for quasilinear elliptic equations involving the p-Laplacian. Electron. J. Differ. Equ., 56: 1–16 (2008)

    MathSciNet  MATH  Google Scholar 

  29. Silva, A.B., Vieira, G.F. Quasilinear asymptotically periodic Schrödinger equations with critical growth. Calc. Var. Partial Differ. Equ., 39: 1–33 (2010)

    Article  MATH  Google Scholar 

  30. Yang, M., Ding, Y. Existence of semiclassical states for a quasilinear Schrödinger equation with critical exponent in RN. Annali di Matematica Pura ed Applicata, 192: 783–804 (2013)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Computer Science, Hubei University of Arts and Science, Xiangyang, 441053, China

    Ji-xiu Wang

Authors
  1. Ji-xiu Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ji-xiu Wang.

Additional information

Supported by the National Natural Science Foundation of China (No. 11501186, 11326145, 11526088).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, Jx. The existence of semiclassical states for some p-Laplacian equation with critical exponent. Acta Math. Appl. Sin. Engl. Ser. 33, 417–434 (2017). https://doi.org/10.1007/s10255-017-0671-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10255-017-0671-4

Keywords

2000 MR Subject Classification

Search

Navigation