Nonlinear Choquard equations: Doubly critical case

Consider nonlinear Choquard equations −Δu+u=(Iα∗F(u))F′(u)in RN,limx→∞u(x)=0,where Iα denotes Riesz potential and α∈(0,N). In this paper, we show that when F is doubly critical, i.e. F(u)=NN+α|u|N+αN+N−2N+α|u|N+αN−2, the nonlinear Choquard equation admits a nontrivial solution if N≥5 and α+4