Skip to main content
SpringerLink
Log in

Positive Solutions to a Neumann Problem of Semilinear Elliptic System with Critical Nonlinearity

  • Original Papers
  • Published:
Acta Mathematicae Applicatae Sinica Aims and scope Submit manuscript

Abstract

In this paper, we consider the Neumann boundary value problem for a system of two elliptic equations involving the critical Sobolev exponents. By means of blowing-up method, we obtain behavior of positives with low energy and asymptotic behavior of positive solutions with minimum energy as the parameters λ, μ→∞.

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. Adimurthi, Mancini, G. The Neumann problem for elliptic equations with critical nonlinearity. A tribute in honor of G. Prodi, Scuola Norm. Sup. Pisa, 9–25 (1991)

  2. Adimurthi, Yadava, S.L. Interaction between the geometry of the boundary and positive solutions of semilinear Neumann problem with critical nonlinearity. J. of Funct. Anal., 113:318–350 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  3. Adimurthi, Yadava, S.L. Existence of nonexistence of positive radial solutions of Neumann problem with critical Sobolev exponents. Arch. Rat. Mech. Anal., 113:275–296 (1991)

    Article  MathSciNet  Google Scholar 

  4. Alves, C.O, de Morais Filho, D.C., Souto, M.A.S. On systems of elliptic equations involving subcritical or critical Sobolev exponents. Nonlinear Anal., 42:771–787 (2000)

    Article  MathSciNet  Google Scholar 

  5. Bahri, A. Critical point at infinity in some variational problems. Pitman Research Notes in Math. Series, Vol. 182, Longman, Green, New York, 1989

  6. Brezis, H., Lieb, E. Relation between pointwise convergence of functions and convergence of functionals. Proc. Amer. Math. Soc., 88:486–490 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  7. Brezis, H., Coron, J.M. Convergence of solution of H-system or how to blow bubbles. Arch. Rat. Mech. Anal., 89:21–56 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  8. Budd, C., Knaap, M.C., Peletier, L.A. Asymptotic behaviour of solutions of elliptic equations with critical exponent and Neumann boundary conditions. Proc. Roy. Soc. Edinburgh, 117A:225–250 (1991)

    MathSciNet  Google Scholar 

  9. Cao, D., Han, P. High energy positive solutions of Neumann Problems for an Elliptic system of equations with critical nonlinearities. Calculus of Variations and Paitial Differential Equations, Calc. Var. 25(2):161–185 (2005)

    Article  Google Scholar 

  10. Chabrowski, J. On multiple solutions of the Neumann problem involving the critical Sobolev exponent. Colloq. Math., 101:203–220 (2004)

    MATH  MathSciNet  Google Scholar 

  11. Chabrowski, J., Willem, M. Least energy solutions of a critical Neumann problem with a weight. Calc. Var. PDE., 15:421–431 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  12. Chabrowski, J., Yang, J. On the neumann problem for an elliptic system of equation involving the sobolev expoent. Colloquium Math., 90:1–35 (2001)

    Article  MathSciNet  Google Scholar 

  13. Grossi, M., F. Pacella, Positive solutions of nonlinear elliptic equations with critical Sobolev exponent and mixed boundary conditions. Proc. Roy. Soc. Edinbergh, 116A:23–43 (1990)

    MathSciNet  Google Scholar 

  14. Keller, E., Segel, A. Initiation of slime mold agrregation viewed as an instability. J. Theoret. Biology, 26:399–425 (1970)

    Article  Google Scholar 

  15. Lin, C.S., Ni, W.M. On the diffusion coefficient of a semilinear neumann problem. Springer Lecture Notes, Vol. 1340, Springer-Verlag, New York, Berlin, 1986

  16. Lin, C.S., Ni, W.M., Takagi, I. Large amplitude stationary solutions to a chemotaxis system. J. Differential Equations, 72:1–27 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  17. Lions, P.L. The concentration compactness principle in the calculus of variations. The limit case (Part 1 and Part 2), Rev. Mat. Iberoamerocana, 1, 145–201, 45–121 (1985)

    MATH  Google Scholar 

  18. Meinhardt, H. Models of Biological Pattern Formation. Academic Press, London, 1982

  19. Ni, W.M., Pan, X.B., Takagi, I. Singular behavior of least energy solutions of a semilinear Neumann problem involving critical Sobolev exponents. Duke Math. J., 67:1–20 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  20. Struwe, M. A global compactness result for elliptic boundary value problems involving limiting nonlinearities. Math. Z., 187:511–517 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  21. Struwe, M. Vatiational methods: Applications to nonlinear partial differential equations and hamiltonian system. Springer-Verlag, Berlin, New York, 1990

  22. Wang, X.J. Neumann problems of semilinear elliptic equations involving critical Sobolev exponents. J. Differential Equations, 93:283–310 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  23. Wang, Z.Q. The effect of the domain geometry on the number of positive solutions of Neumann problems with critical exponents. Diff. Integral Equations, 6:1533–1554 (1995)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Academy of Mathematics and Systems Science, Chinese Academy of Science, Beijing, 100080, China

    Wei-hua Yang

Authors
  1. Wei-hua Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wei-hua Yang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yang, Wh. Positive Solutions to a Neumann Problem of Semilinear Elliptic System with Critical Nonlinearity. Acta Math. Appl. Sin, Engl. Ser. 22, 687–702 (2006). https://doi.org/10.1007/s10255-006-0343-2

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10255-006-0343-2

Keywords

2000 MR Subject Classification

Search

Navigation