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We establish some existence results for the Brezis-Nirenberg type problem ofthe nonlinear Choquard equation $$-\Delta u=\left(\int_{\Omega}\frac{|u|^{2_{\mu}^{\ast}}}{|x-y|^{\mu}}dy\right)|u|^{2_{\mu}^{\ast}-2}u+\lambdau\4.14mm\mbox{in}\1.14mm \Omega, $$ where $\Omega$ is a bounded domain of$\mathbb{R}^N$, with Lipschitz boundary, $\lambda$ is a real parameter,$N\geq3$, $2_{\mu}^{\ast}=(2N-\mu)/(N-2)$ is the critical exponent in the senseof the Hardy-Littlewood-Sobolev inequality.

On the Brezis-Nirenberg type critical problem for nonlinear Choquard equation