Non-radial positive and sign-changing solutions for the FitzHugh–Nagumo system in \(\mathbb {R}^N\)

Abstract

In this article we present the existence of infinitely many non-radial positive or sign-changing solutions for the following FitzHugh–Nagumosystem:

$$\begin{aligned} \left\{ \begin{array}{ll} \Delta u-a(|x|)u+g(u)-\delta v=0, \quad & x\in \mathbb {R}^N,\\ \Delta v+u=0, & x\in \mathbb {R}^N,\\ u(x), ~v(x)\rightarrow 0, & \text{ as }~ |x|\rightarrow +\infty ,\\ \end{array}\right. \end{aligned}$$

where \(N\ge 5\), \(\delta >0\), \(g(u)=(a_0+1)u^2-u^3\), \(0<a_0<\frac{1}{2}\) and \(a(|x|)\in (0,\frac{1}{2})\) satisfies some decay conditions at the infinity. More precisely, for any positive integer k large, there is a \(\delta _k>0\) such that for \(0<\delta <\delta _k\), there exists positive solutions with 2k peaks, which are respectively concentrated at the vertices of a regular k-polygon on two circles in 3-dimensional space with the radium \(r\sim k \ln k\) and the height \(h\sim \frac{1}{k}\). In addition, the sign-changing solutions with 2k peaks are evenly distributed on the equatorial \(\textrm{T}=\{x\in \mathbb {R}^2:x_1^2+x_2^2=r^2\}\) in the \((x_1, x_2)\)-plane. As a by-product, we give the similar results of Schödinger-Poisson in \(\mathbb {R}^N\) for \(N\ge 3\).