On Multiple Radial Solutions of a Singularly Perturbed Nonlinear Elliptic System

We study radial solutions of a singularly perturbed nonlinear elliptic system of the FitzHugh–Nagumo type. In a particular parameter range, we find a large number of layered solutions. First we show the existence of solutions whose layers are well separated from each other and also separated from the origin and the boundary of the domain. Some of these solutions are local minimizers of a related functional while the others are critical points of saddle type. Although the local minimizers may be studied by the Γ‐convergence method, the reduction procedure presented in this paper gives a more unified approach that shows the existence of both local minimizers and saddle points. Critical points of both types are all found in the reduced finite dimensional problem. The reduced finite dimensional problem is solved by a topological degree argument. Next we construct solutions with odd numbers of layers that cluster near the boundary, again using the reduction method. In this case the reduced finite dimensional p...

[1]  Paul C. Fife,et al.  Boundary and interior transition layer phenomena for pairs of second-order differential equations☆ , 1976 .

[2]  L. Modica.  The gradient theory of phase transitions and the minimal interface criterion , 1987 .

[3]  A remark on singular perturbation methods , 1985 .

[4]  Masahisa Tabata,et al.  Multiple Solutions of Two-Point Boundary Value Problems of Neumann Type with a Small Parameter , 1980 .

[5]  Yasumasa Nishiura,et al.  Coexistence of Infinitely Many Stable Solutions to Reaction Diffusion Systems in the Singular Limit , 1994 .

[6]  Xiaofeng Ren,et al.  On energy minimizers of the diblock copolymer problem , 2003 .

[7]  Guido Sweers,et al.  Solutions with internal jump for an autonomous elliptic system of FitzHugh–Nagumo type , 2003 .

[8]  E. N. Dancer,et al.  Multipeak solutions for an elliptic system of Fitzhugh-Nagumo type , 2006 .

[9]  E. N. Dancer,et al.  Multipeak Solutions for the Neumann Problem of an Elliptic System of Fitzhugh–Nagumo Type , 2005 .

[10]  J. NAGUMOt,et al.  An Active Pulse Transmission Line Simulating Nerve Axon , 2006 .

[11]  Yasumasa Nishiura,et al.  Stability of singularly perturbed solutions to systems of reaction-diffusion equations , 1987 .

[12]  E. N. Dancer,et al.  Solutions with interior and boundary peaks for the neumann problem of an elliptic system of fitzhugh-nagumo type , 2006 .

[13]  Shusen Yan,et al.  A minimization problem associated with elliptic systems of FitzHugh–Nagumo type , 2004 .

[14]  G. Klaasen,et al.  Standing wave solutions for a system derived from the Fitzhugh-Nagumo equations for nerve conduction , 1986 .

[15]  Urs Kirchgraber,et al.  Dynamics Reported : Expositions in Dynamical Systems , 1994 .

[16]  E. N. Dancer,et al.  Multi-layer solutions for an elliptic problem☆ , 2003 .

[17]  Xiaofeng Ren,et al.  Concentrically layered energy equilibria of the di-block copolymer problem , 2002, European Journal of Applied Mathematics.

[18]  X. Ren,et al.  Nucleation in the FitzHugh–Nagumo system: Interface–spike solutions , 2005 .

[19]  Robert V. Kohn,et al.  Local minimisers and singular perturbations , 1989, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[20]  R. FitzHugh.  Impulses and Physiological States in Theoretical Models of Nerve Membrane. , 1961, Biophysical journal.

[21]  Stefan Müller,et al.  Singular perturbations as a selection criterion for periodic minimizing sequences , 1993 .

[22]  Xinfu Chen,et al.  Periodicity and Uniqueness of Global Minimizers of an Energy Functional Containing a Long-Range Interaction , 2005, SIAM J. Math. Anal..