Single peak solutions for an elliptic system of FitzHugh–Nagumo type

We study the Dirichlet problem for an elliptic system derived from FitzHugh–Nagummo model as follows:

$$\begin{aligned} \left\{ \begin{aligned}&-\varepsilon ^2\Delta u =f(u)- v, \qquad&\text {in}\ \Omega ,\\&-\Delta v+\gamma v =\delta _\varepsilon u,&\text{ in }\ \Omega ,\\&u=v =0,&\text {on}\ \partial \Omega , \end{aligned} \right. \end{aligned}$$

where \(\Omega \) represents a bounded smooth domain in \(\mathbb {R}^2\) and \(\varepsilon , \gamma \) are positive constants. The parameter \(\delta _{\varepsilon }>0\) is a constant dependent on \(\varepsilon \), and the nonlinear term f(u) is defined as \(u(u-a)(1-u)\). Here, a is a function in \(C^2(\Omega )\cap C^1({\overline{\Omega }})\) with its range confined to \((0,\frac{1}{2})\). Our research focuses on this spatially inhomogeneous scenario whereas the scenario that a is spatially constant has been studied extensively by many other mathematicians. Specifically, in dimension two, we utilize the Lyapunov–Schmidt reduction method to establish the existence of a single interior peak solution. This is contingent upon a mild condition on a, which acts as an indicator of a location-dependent activation threshold for excitable neurons in the biological environment.