Fractional Choquard Equation with Critical Nonlinearities

In this article, we study the Brezis-Nirenberg type problem of nonlinear Choquard equation involving a fractional Laplacian $$(-\text{De}{)}^{s}u=\left({\int }_{\text{Om}}\frac{\mathrm{\mid }u{\mathrm{\mid }}^{2_{\mu ,s}^{\ast }}}{\mathrm{\mid }x-y{\mathrm{\mid }}^{\mu }}\mathrm{d}y\right)\mathrm{\mid }u{\mathrm{\mid }}^{2_{\mu ,s}^{\ast }-2}u+\text{la}u\text{in }\text{Om},$$ where $\text{Om}$ is a bounded domain in ${\mathbb{R}}^n$ with Lipschitz boundary, $\text{la}$ is a real parameter, $s\in (0,1)$, $n>2s$ and $2_{\mu ,s}^{\ast }=(2n-\mu )\mathrm{/}(n-2s)$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality. We obtain some existence, multiplicity, regularity and nonexistence results for solution of the above equation using variational methods.