On Multiple Radial Solutions of a Singularly Perturbed Nonlinear Elliptic System

Abstract

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 problem is solved by a maximization argument.