Existence and nonexistence of nontrivial solution for a Choquard equation with doubly critical exponents in $\mathbb R^N$

This paper deals with the following Choquard equation with doubly critical exponents and a local nonlinear perturbation: \begin{equation*} \begin{cases} - \Delta u+u=\gamma (I_{\alpha}* |u|^{\frac{\alpha}{N}+1} )|u|^{\frac{\alpha}{N}-1}u +\mu |u|^{q-2}u+|u|^{2^*-2}u, & x\in \mathbb R^N; \\ u\in H^1(\mathbb R^N), \end{cases} \end{equation*} where $N\ge 3$, $\alpha\in (0, N)$, $\gamma > 0$, $\mu\ge 0$ and $I_{\alpha}: \mathbb R^N\rightarrow \mathbb R$ is the Riesz potential. The exponent $1+\frac{\alpha}{N}$ is the lower critical with respect to the Hardy-Littlewood-Sobolev inequality. We demonstrate that the aforementioned equation admits no nontrivial solutions when $\mu = 0$; conversely, it possesses a ground state solution under mild conditions on $N, q, \alpha$ and $\gamma$ when $\mu > 0$.