Peak solutions for an elliptic system of FitzHugh-Nagumo type
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Volume 2 (2003) no. 4, p. 679-709
The aim of this paper is to study the existence of various types of peak solutions for an elliptic system of FitzHugh-Nagumo type. We prove that the system has a single peak solution, which concentrates near the boundary of the domain. Under some extra assumptions, we also construct multi-peak solutions with all the peaks near the boundary, and a single peak solution with its peak near an interior point of the domain.
Classification:  35J50,  93C15
@article{ASNSP_2003_5_2_4_679_0,
     author = {Dancer, Edward Norman and Yan, Shusen},
     title = {Peak solutions for an elliptic system of FitzHugh-Nagumo type},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola normale superiore},
     volume = {Ser. 5, 2},
     number = {4},
     year = {2003},
     pages = {679-709},
     zbl = {1115.35039},
     mrnumber = {2040640},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2003_5_2_4_679_0}
}
Dancer, Edward Norman; Yan, Shusen. Peak solutions for an elliptic system of FitzHugh-Nagumo type. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Volume 2 (2003) no. 4, pp. 679-709. http://www.numdam.org/item/ASNSP_2003_5_2_4_679_0/

[1] A. Bahri, “Critical points at infinity in some variational problems", Research Notes in Mathematics 182, Longman-Pitman, 1989. | MR 1019828 | Zbl 0676.58021

[2] P. Clément - G. Sweers, Existence and multiplicity results for a semilinear eigenvalue problem, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 14 (1987), 97-121. | Numdam | MR 937538 | Zbl 0662.35045

[3] E. N. Dancer, A note on asymptotic uniqueness for some nonlinearities which change sign, Bull. Austral. Math. Soc. 61 (2000), 305-312. | MR 1748710 | Zbl 0945.35031

[4] E. N. Dancer - S. Yan, Interior and boundary peak solutions for a mixed boundary value problem, Indiana Univ. Math. J. 48 (1999), 1177-1212. | MR 1757072 | Zbl 0948.35055

[5] E. N. Dancer - S. Yan, Singularly perturbed elliptic problem in exterior domains, J. Differential Integral Equations 13 (2000), 747-777. | MR 1750049 | Zbl 1038.35008

[6] E. N. Dancer - S. Yan, A minimization problem associated with elliptic system of FitzHugh-Nagumo type, Ann. Inst. H. Poincaré, Analyse Non Linéaire, to appear. | Numdam | MR 2047356 | Zbl 1110.35019

[7] D. G. deFigueiredo - E. Mitidieri, A maximum principle for an elliptic system and applications to semilinear problems, SIAM J. Math. Anal. 17 (1986), 836-849. | MR 846392 | Zbl 0608.35022

[8] R. Fitzhugh, Impulses and physiological states in theoretical models of nerve membrane, Biophy. J. 1 (1961), 445-466.

[9] R. Gardner - L. A. Peletier, The set of positive solutions of semilinear equations in large ball, Proc. Royal Soc. Edinburgh 104A (1986), 53-72. | MR 877892 | Zbl 0625.35030

[10] A. Hodgkin - A. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol. 117 (1952), 500-544.

[11] G. A. Klaasen - E. Mitidieri, Standing wave solutions for a system derived from the FitzHugh-Nagumo equation for nerve conduction, SIAM J. Math. Anal. 17 (1986), 74-83. | MR 819214 | Zbl 0593.35043

[12] G. A. Klaasen - W. C. Troy, Stationary wave solutions of a system of reaction-diffusion equations derived from the FitzHugh-Nagumo equations, SIAM J. Appl. Math. 44 (1984), 96-110. | MR 730003 | Zbl 0543.35051

[13] J. S. Nagumo - S. Arimoto - Y. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. Inst. Radio. Engineers 50 (1962), 2061-2070.

[14] W. M. Ni - I. Takagi - J. Wei, On the location and profile of spike-layer solutions to a singularly perturbed semilinear Dirichlet problem, intermediate solution, Duke Math. J. 94 (1998), 597-618. | MR 1639546 | Zbl 0946.35007

[15] W. M. Ni - J. Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problem, Comm. Pure Appl. Math. 48 (1995), 731-768. | MR 1342381 | Zbl 0838.35009

[16] L. A. Peletier - J. Serrin, Uniqueness of positive solutions of semilinear equations in n , Arch. Rat. Mech. Anal. 81 (1983), 181-197. | MR 682268 | Zbl 0516.35031

[17] C. Reinecke - G. Sweers, A boundary layer solution to a semilinear elliptic system of FitzHugh-Nagumo type, C. R. Acad. Sci. Paris Sér. I. Math. 329 (1999), 27-32. | MR 1703287 | Zbl 0931.35046

[18] C. Reinecke - G. Sweers, A positive solution on N to a system of elliptic equations of FitzHugh-Nagumo type, J. Differential Equations 153 (1999), 292-312. | MR 1683624 | Zbl 0929.35042

[19] C. Reinecke - G. Sweers, Existence and uniqueness of solutions on bounded domains to a FitzHugh-Nagumo type elliptic system, Pacific J. Math. 197 (2001), 183-211. | MR 1810215 | Zbl 1066.35069

[20] C. Reinecke - G. Sweers, Solutions with internal jump for an autonomous elliptic system of FitzHugh-Nagumo type, Math. Nachr. 251 (2003), 64-87. | MR 1960805 | Zbl 1118.35009

[21] O. Rey, The role of the Green's function in a non-linear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal. 89 (1990), 1-52. | MR 1040954 | Zbl 0786.35059

[22] G. Sweers - W. Troy, On the bifurcation curve for an elliptic system of FitzHugh-Nagumo type, Physica D: Nonlinear Phenomena 177 (2003), 1-22. | MR 1965323 | Zbl 1082.35068