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: \(\begin{aligned} \left\{ \begin{aligned}&-\varepsilon ^2\Delta u =f(u)- v, \qquad&\text {in}\ \Omega ,\\&-\Delta v+\gamma v =\delta _\varepsilon u,&\text{ in }\ \Omega ,\\&u=v =0,&\text {on}\ \partial \Omega , \end{aligned} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{ccc}& -{\epsilon }^2\mathrm{\Delta }u=f(u)-v,\hfill & \hfill \text{in}\mathrm{\Omega },\\ \hfill & -\mathrm{\Delta }v+\gamma v={\delta }_{\epsilon }u,\hfill & \hfill \text{in}\mathrm{\Omega },\\ \hfill & u=v=0,\hfill & \hfill \text{on}\partial \mathrm{\Omega },\end{array}\right)\end{array}$$ where \(\Omega \) $\mathrm{\Omega }$ represents a bounded smooth domain in \(\mathbb {R}^2\) ${\mathbb{R}}^2$ and \(\varepsilon , \gamma \) $\epsilon ,\gamma$ are positive constants. The parameter \(\delta _{\varepsilon }>0\) ${\delta }_{\epsilon }>0$ is a constant dependent on \(\varepsilon \) $\epsilon$ , and the nonlinear term f(u) is defined as \(u(u-a)(1-u)\) $u(u-a)(1-u)$ . Here, a is a function in \(C^2(\Omega )\cap C^1({\overline{\Omega }})\) $C^2(\mathrm{\Omega })\cap C^1(\overline{\mathrm{\Omega }})$ with its range confined to \((0,\frac{1}{2})\) $(0,\frac{1}{2})$ . 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.