A variational approach for standing waves of FitzHugh–Nagumo type systems

We study the existence of radially symmetric solutions of FitzHugh–Nagumo type elliptic systems in RNRN (N⩾2N⩾2):equation(⁎)−Δu=g(u)−vin RN,−dΔv+γv=uin RN,(u(x),v(x))→(0,0)as |x|→∞. We utilize a truncation technique and apply minimax arguments to the corresponding strongly indefinite functionalIγ(u,v)=12∫RN|∇u|2−d|∇v|2dx−∫RNG(u)+γ2v2−uvdx, defined on Hr1(RN)×Hr1(RN), to find positive and possibly sign-changing solutions of (⁎). In particular, we overcome difficulty related to Palais–Smale condition via our new scaling argument. When g(ξ)=ξ(1−ξ)(ξ−α)g(ξ)=ξ(1−ξ)(ξ−α), α∈(0,12), we improve the existence result of Reinecke–Sweers [23].