In this article, we consider the fractional Choquard equation involving a fractional Laplacian \( (-\Delta )^{s} u=\left( \int \limits _{\mathbb {R}^N} \frac{|u(y)|^{2_{\mu ,s}^*}}{|x-y|^\mu } dy\right) \vert u\vert ^{2_{\mu ,s}^*-2} u \pm \vert u\vert ^{q-2} u \quad \text{ in } \quad \mathbb {R}^N, \) where \(s\in (0,1), 1<q \le 2_{\mu ,s}^*, 2_{\mu ,s}^*=\frac{2 N-\mu }{N-2s}\) and \(N >2s \) . By the Pohozaev type identity, we prove the nonexistence of solutions for \(1<q<2_{s}^*\) . In the case of double critical exponents, i.e. \(q=2_{s}^*\) , we use the Nehari manifold and Mountain Pass theorem, to prove the existence of radial ground state solutions.
