Skip to main content
SpringerLink
Log in

Positive Solutions for a Quasilinear Schrödinger Equation with Critical Growth

  • Published:
Journal of Dynamics and Differential Equations Aims and scope Submit manuscript

Abstract

We consider the quasilinear problem

$$-\varepsilon^p\text{div}(|\nabla u|^{p-2}\nabla u) + V(z)u^{p-1} = f(u) + u^{p^*-1},\,u \in W^{1,p}\left(\mathbb{R}^N\right), $$

where ε > 0 is a small parameter, 1 < p < N, p* = Np/(N − p), V is a positive potential and f is a superlinear function. Under a local condition for V we relate the number of positive solutions with the topology of the set where V attains its minimum. In the proof we apply Ljusternik-Schnirelmann theory.

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. Alves C.O.: Existence and multiplicity of solutions for a class of quasilinear equations. Adv. Non. Studies 5, 73–87 (2005)

    MathSciNet  MATH  Google Scholar 

  2. Alves C.O., Figueiredo G.M.: Multiplicity of positive solutions for a quasilinear problem in \({\mathbb{R}^N}\) via penalization method. Adv. Non. Stud. 5, 551–572 (2005)

    MathSciNet  MATH  Google Scholar 

  3. Alves C.O., do Ó J.M., Souto M.A.S.: Local mountain-pass for a class of elliptic problems in \({\mathbb R^N}\) involving critical growth. Nonlinear Anal. 46, 495–510 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ambrosetti A., Malchiodi A., Secchi S.: Multiplicity results for some nonlinear Schrödinger equations with potentials. Arch. Rational Mech. Anal. 159, 253–271 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  5. Benci V., Cerami G.: The effect of the domain topology on the number of positive solutions of nonlinear elliptic problems. Arch. Rational Mech. Anal. 114, 79–93 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  6. Benci V., Cerami G.: Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology. Calc. Var. Partial Differ. Equ. 2, 29–48 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  7. Brezis H., Nirenberg L.: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Comm. Pure Appl. Math. 36, 437–477 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cingolani S., Lazzo M.: Multiple semiclassical standing waves for a class of nonlinear Schrödinger equations. Topol. Methods Nonlinear Anal. 10, 1–13 (1997)

    MathSciNet  MATH  Google Scholar 

  9. Cingolani S., Lazzo M.: Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions. J. Differ. Equ. 160, 118–138 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  10. Del Pino M., Felmer P.: Local Mountain Pass for semilinear elliptic problems in unbounded domains. Calc. Var. Partial Differ. Equ. 4, 121–137 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  11. Di Benedetto E.: C 1,α local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal. 7, 827–850 (1983)

    Article  MathSciNet  Google Scholar 

  12. Figueiredo G.M.: Existence, multiplicity and concentration of positive solutions for a class of quasilinear problems with critical growth. Comm. Appl. Nonlinear Anal. 13, 79–99 (2006)

    MathSciNet  MATH  Google Scholar 

  13. Floer A., Weinstein A.: Nonspreading wave packets for the cubic Schrdinger equation with a bounded potential. J. Funct. Anal. 69, 397–408 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ghoussoub N.: Duality and pertubation methods in critical point theory. Cambridge University Press, Cambridge (1993)

    Book  Google Scholar 

  15. Lazzo, M.: Existence and multiplicity results for a class of nonlinear elliptic problems on \({\mathbb{R}^N}\) . Discret. Contin. Dynam. Systems, suppl., 526–535 (2003)

  16. Li G.B.: Some properties of weak solutions of nonlinear scalar fields equations. Ann. Acad. Sci. Fenn. Ser. A I Math. 15, 27–36 (1990)

    MathSciNet  MATH  Google Scholar 

  17. Lions P.L.: The concentration-compactness principle in the calculus of variations. The limit case. I. Rev. Mat. Iberoam. 1, 145–201 (1985)

    Article  MATH  Google Scholar 

  18. Miyagaki O.H.: On a class of semilinear elliptic problems in \({\mathbb{R}^{N} }\) with critical growth. Nonlinear Anal. 29, 773–781 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  19. do Ó J.M.: On existence and concentration of positive bound states of p-Laplacian equation in \({\mathbb{R}^N}\) involving critical growth. Nonlinear Anal. 62, 777–801 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. Oh Y.-G.: Existence of semiclassical bound states of nonlinear Schrdinger equations with potentials of the class (V) a . Comm. Partial Differ. Equ. 13, 1499–1519 (1998)

    Article  Google Scholar 

  21. Oh Y.-G.: Correction to: “Existence of semiclassical bound states of nonlinear Schrdinger equations with potentials of the class (V) a ”. Comm. Partial Differ. Equ. 14, 833–834 (1989)

    MATH  Google Scholar 

  22. Pomponio A., Secchi S.: On a class of singularly perturbed elliptic equations in divergence form: existence and multiplicity results. J. Differ. Equ. 207, 228–266 (2004)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  24. Trudinger N.S.: On Harnack type inequalities and their application to quasilinear elliptic equations. Comm. Pure Appl. Math. 20, 721–747 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  25. Wang X.: On concentration of positive bound states of nonlinear Schrdinger equations. Comm. Math. Phys. 153, 229–244 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  26. Willem M.: Minimax Theorems. Birkhäuser, Basel (1996)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemática, Universidade Federal do Pará, Belém, PA, 66075-100, Brazil

    Giovany M. Figueiredo

  2. Departamento de Matemática, Universidade de Brasília, Brasília, DF, 70910-900, Brazil

    Marcelo F. Furtado

Authors
  1. Giovany M. Figueiredo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marcelo F. Furtado
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Giovany M. Figueiredo.

Additional information

The article is published in memory of Prof. Jack Hale.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Figueiredo, G.M., Furtado, M.F. Positive Solutions for a Quasilinear Schrödinger Equation with Critical Growth. J Dyn Diff Equat 24, 13–28 (2012). https://doi.org/10.1007/s10884-011-9231-4

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10884-011-9231-4

Keywords

Mathematics Subject Classification (2000)

Search

Navigation