Nonlinear Choquard equations: doubly critical case

Consider nonlinear Choquard equations \begin{equation*} \left{\begin{array}{rcl} -\Delta u +u & = &(I_\alphaF(u))F’(u) \quad \text{in } $\mathbb{R}$^N, \ \lim_{x \to \infty}u(x) & = &0, \end{array}\right. \end{equation} where $I_{\alpha }$ denotes Riesz potential and $\alpha \in (0, N)$\mathrm{.}Inthispaper,weshowthatwhen$F$isdoublycritical,i\mathrm{.}e\mathrm{.}$F(u) = \frac{N}{N+\alpha}|u|^{\frac{N+\alpha}{N}}+\frac{N-2}{N+\alpha}|u|^{\frac{N+\alpha}{N-2}}$, the nonlinear Choquard equation admits a nontrivial solution if $N\ge 5$ and $\alpha +4<N$.