Skip to main content
SpringerLink
Log in

Existence and concentration of positive ground states for a Kirchhoff equation involving critical Sobolev exponent

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

In this paper, we consider the following semilinear Kirchhoff type equation

$$\left\{\begin{array}{ll}-\left(\epsilon^2a + \epsilon b \int \limits_{\mathbb{R}^3}|\nabla u|^2 \right) \triangle {u}+V(x)u = \mu K(x)|u|^{p-1}u + Q(x)|u|^4u, \,\, \mathrm{in}\, \mathbb{R}^3, \\ u \in H^1(\mathbb{R}^3), \,\, u > 0,\end{array}\right.$$

where \({\epsilon > 0}\) is a small parameter, \({p \in [3,5)}\), a, b are positive constants, μ > 0 is a parameter, and the nonlinear growth of |u|4 u reaches the Sobolev critical exponent since 2* = 6 for three spatial dimensions. We prove the existence of a positive ground state solution \({u_\epsilon}\) with exponential decay at infinity for μ > 0 and \({\epsilon}\) sufficiently small under some suitable conditions on the nonnegative functions V, K and Q. Moreover, \({u_\epsilon}\) concentrates around a global minimum point of V as \({\epsilon \rightarrow 0^+}\). The methods used here are based on the concentration-compactness principle of Lions.

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.

Similar content being viewed by others

References

  1. Alves C., Corrêa F., Ma T.: Positive solutions for a quasilinear elliptic equation of Kirchhoff type. Comput. Math. Appl. 49, 85–93 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  2. Alves C., Figueiredo G.: Nonliear perturbations of a periodic Krichhoff equation in \({\mathbb{R}^{N}}\). Nonlinear Anal. 75, 2750–2759 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  3. Alves C., Corrêa F., Figueiredo G.: On a class of nonlocal elliptic problems with critical growth. Differ. Equ. Appl. 2, 409–417 (2010)

    MATH  MathSciNet  Google Scholar 

  4. Arosio A., Panizzi S.: On the well-posedness of the Kirchhoff string. Trans. Am. Math. Soc. 348, 305–330 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  5. Azzollini A.: The elliptic Kirchhoff equation in \({\mathbb{R}^N}\) perturbed by a local nonlinearity. Differ. Integr. Equ. 25, 543–554 (2012)

    MATH  MathSciNet  Google Scholar 

  6. Brezis H., Kato T.: Remarks on the Schrodinger operator with regular complex potentials. J. Math. Pures Appl. 58, 137–151 (1979)

    MATH  MathSciNet  Google Scholar 

  7. Brezis H., Lieb EH.: A relation between pointwise convergence of functions and convergence of functionals. Proc. Am. Math. Soc. 8, 486–490 (1983)

    Article  MathSciNet  Google Scholar 

  8. Brezis H., Nirenberg L.: Positive solutions of nonlinear elliptic problems involving critical Sobolev exponent. Commun. Pure Appl. Math. 36, 437–477 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  9. Chen C., Kuo Y., Wu T.: The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions. J. Differ. Equ. 250, 1876–1908 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  10. Cheng B.: New existence and multiplicity of nontrivial solutions for nonlocal elliptic Kirchhoff type problems. J. Math. Anal. Appl. 394, 488–495 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  11. Chipot M., Lovat B.: Some remarks on non local elliptic and parabolic problems. Nonlinear Anal. 30, 4619–4627 (1997)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  13. Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order, 2nd ed., Grundlehren Math. Wiss., 224 Springer, Berlin (1983)

  14. Ding W., Ni W.: On the existence of positive entire solutions of a semilinear elliptic equation. Arch. Ration. Mech. Anal. 91, 283–308 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  15. Ekeland I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  16. Figueiredo G., Junior J.: Multiplicity of solutions for a Kirchhoff equation with subcritical or critical growth. Differ. Integ. Equ. 25, 853–868 (2012)

    MATH  Google Scholar 

  17. Figueiredo G.: Existence of a positive solution for a Kirchhoff problem type with critical growth via truncation argument. J. Math. Anal. Appl. 401, 706–713 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  18. Figueiredo G.: Multiplicity and concentration behavior of positive solutions for a Schrödinger-Kirchhoff type problem via penalization method, ESAIM control. Optim. Calc. Var. 20, 389–415 (2014)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  20. He X., Zou W.: Existence and concentration behavior of positive solutions for a Kirchhoff equation in R 3. J. Differ. Equ. 252, 1813–1834 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  21. He X., Zou W.: Existence and concentration of ground states for Schrödinger-Poisson equations with critical growth. J. Math. Phys. 53, 023702 (2012). doi:10.1063/1.3683156

    Article  MathSciNet  Google Scholar 

  22. He., Y., Li., G., Peng, S.: Concentrating Bound States for Kirchhoff Type Problems in \({\mathbb{R}^3}\) Involving Critical Sobolev Exponents, arXiv:1306.0122 [math.AP]

  23. Huang L., Rocha E., Chen J.: Positive and sign-changing solutions of a Schrödinger-Poisson system involving a critical nonlinearity. J. Math. Anal. Appl. 1, 55–69 (2013)

    Article  MathSciNet  Google Scholar 

  24. Jin J., Wu X.: Infinitely many radial solutions for Kirchhoff-type problems in R N. J. Math. Anal. Appl. 369, 564–574 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  25. Kirchhoff., G.: Mechanik, Teubner, Leipzig (1883)

  26. Li G.: Some properties of weak solutions of nonlinear scalar field equations. Ann. Acad. Sci. Fenn. AI Math. 15, 27–36 (1990)

    MATH  Google Scholar 

  27. Li, G., Ye, H.: Existence of Positive Solutions For Nonlinear Kirchhoff Type Problems in \({\mathbb{R}^3}\) With Critical Sobolev Exponent and Sign-Changing Nonlinearities, arXiv:1305.6777 [math.AP]

  28. Li, G., Ye, H.: Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in \({\mathbb{R}^3}\). J. Differ. Equ. doi:10.1016/j.jde.2014.04.011

  29. Li Y., Li F., Shi J.: Existence of a positive solution to Kirchhoff type problems without compactness conditions. J. Differ. Equ. 253, 2285–2294 (2012)

    Article  MATH  Google Scholar 

  30. Liang S., Shi S.: Soliton solutions to Kirchhoff type problems involving the critical growth in \({\mathbb{R}^N}\). Nonlinear Anal. 81, 31–41 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  31. Liang, Z., Li, F., Shi, J.: Positive Solutions to Kirchhoff Type Equations With Nonlinearity Having Prescribed Asymptotic Behavior. Ann.I.H.Poincaré-AN. http://dx.doi.org/10.1016/j.anihpc.2013.01.006

  32. Lions, J.: On some questions in boundary value problems of mathematical physics. In: Contemporary Developments in Continuum Mechanics and Partial Differential Equations. Proc. Internat. Sympos. Inst. Mat. Univ. Fed. Rio de Janeiro, (1977)

  33. Lions, J.: On some questions in boundary value problems of mathematical physics. In: North-Holland Math. Stud., vol. 30, pp. 284–346 (1978)

  34. Lions, P.: The Concentration Compactness Principle in the Calculus of Variations: The Locally Compact Case. Parts 1,2. In: Ann. Inst. H. Poincar Anal. Non Linaire, vol. 1, pp. 109–145 (1984a)

  35. Lions, P.: The Concentration Compactness Principle in the Calculus of Variations: The Locally Compact Case. Parts 1,2. In: Ann. Inst. H. Poincar Anal. Non Linaire, vol. 2, pp. 223–283 (1984b)

  36. Liu Z., Guo S.: Positive solutions for asymptotically linear Schrödinger-Kirchhoff-type equations. Math. Meth. Appl. Sci. 37, 571–580 (2014)

    Article  MATH  Google Scholar 

  37. Ma T., Rivera J.: Positive solutions for a nonlinear nonlocal elliptic transmission problem. Appl. Math. Lett. 16, 243–248 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  38. Mao A., Zhang Z.: Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition. Nonlinear Anal. 70, 1275–1287 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  39. Nie J., Wu X.: Existence and multiplicity of non-trivial solutions for Schrödinger-Kirchhoff-type equations with radial potential. Nonlinear Anal. 75, 3470–3479 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  40. Perera K., Zhang Z.: Nontrivial solutions of Kirchhoff-type problems via the Yang index. J. Differ. Equ. 221, 246–255 (2006)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  42. Wang J., Tian L., Xu J., Zhang F.: Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth. J. Differ. Equ. 253, 2314–2351 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  43. Wang X.: On concentration of positive bound states of nonlinear Schrödinger equations. Commun. Math. Phys. 153, 229–244 (1993)

    Article  MATH  Google Scholar 

  44. Wang X., Zeng B.: On concentration of positive bound states of nonlinear Schrödinger equations with competing potential functions. SIAM J. Math. Anal. 28, 633–655 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  45. Wu X.: Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in \({\mathbb{R}^N}\). Nonlinear Anal. RWA. 12, 1278–1287 (2011)

    Article  MATH  Google Scholar 

  46. Zhang Z., Perera K.: Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow. J. Math. Anal. Appl. 317, 456–463 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  47. Zhao L., Zhao F.: Positive solutions for the Schrödinger-Poisson equations with a critical exponent. Nonlinear Anal. 70, 2150–2164 (2009)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Mathematics and Econometrics, Hunan University, Changsha, 410082, Hunan, People’s Republic of China

    Zhisu Liu & Shangjiang Guo

Authors
  1. Zhisu Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shangjiang Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhisu Liu.

Additional information

Research supported by the NSFC (Grant Nos. 11271115), by the Doctoral Fund of Ministry of Education of China, and the Hunan Provincial Natural Science Foundation (Grant No. 14JJ1025).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, Z., Guo, S. Existence and concentration of positive ground states for a Kirchhoff equation involving critical Sobolev exponent. Z. Angew. Math. Phys. 66, 747–769 (2015). https://doi.org/10.1007/s00033-014-0431-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00033-014-0431-8

Mathematics Subject Classification (2010)

Keywords

Search

Navigation