Existence of solutions for critical Fractional FitzHugh–Nagumo type systems

In this paper we study the existence of radially symmetric solutions for a Fractional FitzHugh–Nagumo type systems 0.1 (−Δ)su+u=f(u)−vinRN,(−Δ)sv+v=uinRN,\begin{equation} \def\eqcellsep{&}\begin{array}{rcll} (-\Delta )^su + u & = & f(u) - v & \mbox{in}\;\;\mathbb {R}^N, \\[3pt] (-\Delta )^sv+v & = & u & \mbox{in}\;\;\mathbb {R}^N, \end{array} \end{equation}where s∈(0,1)$s\in (0,1)$, N>2s$N> 2s$, (−Δ)s$(-\Delta )^s$ denotes the fractional Laplacian operator and f:R→R$f:\mathbb {R} \rightarrow \mathbb {R}$ is a continuous function which is allowed to have critical growth: polynomial in case N≥2$N\ge 2$ and exponential if N=1$N=1$ and s=1/2$s={1}/{2}$. We transform the system into an equation with a nonlocal term. We find a critical point of the corresponding energy functional defined in the space of functions with norm endowed by a scalar product that takes into account such nonlocal term. For that matter, and due to the lack of compactness, we deal with weak convergent minimizing sequences and sequences of Lagrange multipliers of an action minima problem.