Asymptotic profiles of variational solutions for a FitzHugh-Nagumo-type elliptic system

Abstract

In this paper we consider a semilinear elliptic system of FitzHugh-Nagumo type on a bounded domain with the same diffusion constant $\lambda^{-1}$ under the Dirichlet boundary condition $-\Delta u=\lambda(f(u)-v)$, $-\Delta v=\lambda(\delta u-\gamma v)$ in $\Omega$. In some parameter range on $(\delta, \gamma)$ this system has at least two nontrivial solutions when $\lambda$ is sufficiently large, and these solutions are obtained by variational methods. We study the asymptotic profiles of these solutions for large $\lambda$ in some parameter range on $(\delta, \gamma)$, especially for small $\delta$ and large $\gamma$.