Standing Waves in the FitzHugh-Nagumo System and a Problem in Combinatorial Geometry

We show that there is a close relation between standing-wave solutions for the FitzHugh-Nagumo system \[ \Delta u +u(u-a)(1-u) - \delta v=0, \ \ \Delta v-\delta \gamma v + u=0 \ \ \mbox{in} \ R^N,\] \[ u, v \to 0 \ \mbox{as} \ |x| \to +\infty \] where $0<a<1/2$ and $\delta \gamma=\beta^2 \in (0, a)$, and the following combinatorial problem: {\it $ (*) \ \ \ $ Given $K$ points $Q_1, ..., Q_K \in R^N$ with minimum distance $1$, find out the maximum number of times that the minimum distance $1$ can occur. } More precisely, we show that for any given positive integer $K$, there exists a $\delta_{K}>0$ such that for $0<\delta <\delta_K$, there exists a standing-wave solution $(u_{\delta},v_{\delta})$ to the FitzHugh-Nagumo system with the property that $u_{\delta}$ has $K$ spikes $Q_{1}^\delta, ..., Q_K^\delta$ and $ (\frac{1}{l^\delta} Q_1^\delta, ..., \frac{1}{l^\delta} Q_K^\delta)$ approaches an optimal configuration in (*), where $l^\delta=\min_{i \not = j} |Q_i^\delta -Q_j^\delta| = \frac{1}{ \sqrt{a} -\beta} \log \frac{1}{\delta} ( 1+o(1))$