On Multiple Radial Solutions of a Singularly Perturbed Nonlinear Elliptic System

We study radial solutions of a singularly perturbed nonlinear elliptic system of the FitzHughNagumo 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 an odd number 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. Supported in part by ARC. Phone: 61 2 9351 4141. Fax: 61 2 9351 4534. E-mail: [email protected] Supported in part by NSF grant DMS-0509725. Phone: 1 (435) 797-0755. Fax: 1 (435) 797-1822. E-mail: [email protected] Phone: 61 2 6773 3154. Fax: 61 2 6773 3312. E-mail: [email protected] 1 Key workds. Radial solutions. Singularly perturbed elliptic system, Lyapunov-Schmidt reduction. Γ-convergence. AMS subject classifications. 35J50, 35J55. Short title. Multiple radial solutions.