Skip to main content
SpringerLink
Log in

Multipeak solutions for an elliptic system of Fitzhugh-Nagumo type

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

We consider the Dirichlet problem for an elliptic system of FitzHugh-Nagumo type. We prove that the problem has solutions with arbitrary many sharp peaks near the boundary but not on the boundary.

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. Bahri, A.: Critical points at infinity in some variational problems. Research Notes in Mathematics 182, Longman-Pitman, 1989

  2. Chang, K.C.: Infinite dimensional Morse theory and multiple solution problems. Birkhauser, Boston, 1993

  3. Clément, P., Sweers, G.: Existence and multiplicity results for a semilinear eigenvalue problem. Ann. Scuola Norm. Sup. Pisa 14, 97–121 (1987)

    MATH  Google Scholar 

  4. Dancer, E.N.: A note on asymptotic uniqueness for some nonlinearities which change sign. Bull. Austral. Math. Soc. 61, 305–312 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  5. Dancer, E.N., Yan, S.: A singularly perturbed elliptic problem in bounded domains with nontrivial topology. Adv. Diff. Equations 4, 347–368 (1999)

    MATH  MathSciNet  Google Scholar 

  6. Dancer, E.N., Yan, S.: Effect of the domain geometry on the existence of multipeak solutions for an elliptic problem. Top. Meth. Nonlinear Anal. 14, 1–38 (1999)

    MATH  MathSciNet  Google Scholar 

  7. Dancer, E.N., Yan, S.: Interior and boundary peak solutions for a mixed boundary value problem. Indiana Uni. Math. J. 48, 1177–1212 (1999)

    MATH  MathSciNet  Google Scholar 

  8. Dancer, E.N., Yan, S.: Singularly perturbed elliptic problem in exterior domains. J. Diff. Int. Equations 13, 747–777 (2000)

    MATH  MathSciNet  Google Scholar 

  9. Dancer, E.N., Yan, S.: A minimization problem associated with elliptic system of FitzHugh-Nagumo type. Ann. Inst. H. Poincaré, Analyse Non Linéaire 21, 237–257 (2004)

    MATH  MathSciNet  Google Scholar 

  10. Dancer, E.N., Yan, S.: Peak solutions for an elliptic system of FitzHugh-Nagumo type. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 2, 679–709 (2003)

    MathSciNet  MATH  Google Scholar 

  11. Dancer, E.N., Yan, S.: Peak solutions for the Neumann problem of an elliptic system of FitzHugh-Nagumo type. Proc. London Math. Soc. 90, 209–244 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  12. Dancer, E.N., Yan, S.: Solutions with interior and boundary peaks for the Neumann problem of an elliptic system of FitzHugh-Nagumo type. Indiana Univ. Math. J., 55, 217–258 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. deFigueiredo, D.G., Mitidieri, E.: A maximum principle for an elliptic system and applications to semilinear problems. SIAM J. Math. Anal. 17, 836–849 (1986)

    Article  MathSciNet  Google Scholar 

  14. Dold, A.: Lectures on Algebraic Topology. Springer-Verlag, Berlin, Heidelberg, New York, 1972

  15. FitzHugh, R.: Impulses and physiological states in theoretical models of nerve membrane. Biophy. J. 1, 445–466 (1961)

    Article  Google Scholar 

  16. Gardner, R., Peletier, L.A.: The set of positive solutions of semilinear equations in large ball. Proc. Royal Soc. Edinburgh 104A, 53–72 (1986)

    MATH  Google Scholar 

  17. Gui, C., Wei, J., Winter, M.: Multiple boundary peak solutions for some singularly perturbed Neumann problems. Ann. Inst. H. Poincaré Anal. Non Linéaire 17, 47–82 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  18. Hodgkin, A., Huxley, A.: A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117, 500–544 (1952)

    Google Scholar 

  19. Klaasen, G.A., Mitidieri, E.: Standing wave solutions for a system derived from the FitzHugh-Nagumo equation for nerve conduction. SIAM J. Math. Anal. 17, 74–83 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  20. Klaasen, G.A., Troy, W.C.: Stationary wave solutions of a system of reaction-diffusion equations derived from the FitzHugh-Nagumo equations. SIAM J. Appl. Math. 44, 96–110 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  21. Nagumo, J.S., Arimoto, S., Yoshizawa, Y.: An active pulse transmission line simulating nerve axon. Proc. Inst. Radio. Engineers 50, 2061–2070 (1962)

    Google Scholar 

  22. Ni, W.M., Takagi, I., Wei, J.: On the location and profile of spike-layer solutions to a singularly perturbed semilinear Dirichlet problem, intermediate solution. Duke Math. J. 94, 597–618 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  23. Ni, W.M., Wei, J.: On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problem. Comm. Pure Appl. Math. 48, 731–768 (1995)

    MATH  MathSciNet  Google Scholar 

  24. Oshita, Y.: On the stable nonconstant stationary solutions and mesoscopic patterns for FitzHugh-Nagumo equations in higher dimensions. J. Diff. Equations 188, 110–134 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  25. Peletier, L.A., Serrin, J.: Uniqueness of positive solutions of semilinear equations in R n. Arch. Rat. Mech. Anal. 81, 181–197 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  26. Reinecke, C., Sweers, G.: A boundary layer solution to a semilinear elliptic system of FitzHugh-Nagumo type. C. R. Acad. Sci. Paris Sér. I. Math. 329, 27–32 (1999)

    MATH  MathSciNet  Google Scholar 

  27. Reinecke, C., Sweers, G.: A positive solution on R N to s system of elliptic equations of FitzHugh-Nagumo type. J. Diff. Equations 153, 292–312 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  28. Reinecke, C., Sweers, G.: Existence and uniqueness of solutions on bounded domains to a FitzHugh-Nagumo type elliptic system. Pacific J. Math. 197, 183–211 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  29. Reinecke, C., Sweers, G.: Solutions with internal jump for an autonomous elliptic system of FitzHugh-Nagumo type. Math. Nachrichten 251, 64–87 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  30. Ren, X., Wei, J.: On the energy minimizers of the diblock copolymer problem. Interfaces Free Bound. 5, 193–238 (2003)

    MATH  MathSciNet  Google Scholar 

  31. Rey, O.: The role of the Green's function in a non-linear elliptic equation involving the critical Sobolev exponent. J. Funct. Anal. 89, 1–52 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  32. Sweers, G., Troy, W.: On the bifurcation curve for an elliptic system of FitzHugh-Nagumo type. Physica D: Nonlinear Phenomena 177, 1–22 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  33. Wei, J.: Conditions for two-peaked solutions of singularly perturbed elliptic equations. Manuscripta Math. 96, 113–131 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  34. Wei, J., Winter, W.: Clustered spots in the FitzHugh-Nagumo system. J. Differential Equations 213, 121–145 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  35. Yan, S.: On the number of interior multipeak solutions for singularly perturbed Neumann problems. Topological Methods in Nonlinear Anal. 12, 61–78 (1999)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, University of Sydney, NSW, 2006, Australia

    E.N. Dancer

  2. School of Mathematics, Statistics and Computer Science, The University of New England, Armidale, NSW, 2351, Australia

    Shusen Yan

Authors
  1. E.N. Dancer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shusen Yan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shusen Yan.

Additional information

The work of the first author was partially supported by the ARC

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dancer, E., Yan, S. Multipeak solutions for an elliptic system of Fitzhugh-Nagumo type. Math. Ann. 335, 527–569 (2006). https://doi.org/10.1007/s00208-006-0754-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-006-0754-7

Mathematics Subject Classification (2000)

Search

Navigation