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

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.

我们研究源自 FitzHugh-Nagummo 模型的椭圆系统的狄利克雷问题，如下所示：

其中\(\Omega \)表示\(\mathbb {R}^2\)中的有界平滑域，\(\varepsilon , \gamma \)是正常数。参数\(\delta _{\varepsilon }>0\)是一个依赖于\(\varepsilon \)的常数，非线性项f ( u ) 定义为\(u(ua)(1-u)\ ）。这里，a是\(C^2(\Omega )\cap C^1({\overline{\Omega }})\)中的函数，其范围限制为\((0,\frac{1}{2 })\)。我们的研究重点是这种空间不均匀的情况，而a是空间常数的情况已被许多其他数学家广泛研究。具体来说，在二维中，我们利用 Lyapunov-Schmidt 约简方法来确定单内峰解的存在性。这取决于a的温和条件，它充当生物环境中可兴奋神经元的位置依赖性激活阈值的指标。