In this paper, we consider a critical Choquard-type Brezis-Nirenberg problem in a smooth bounded domain \(\Omega \subset \mathbb {R}^3\) , which extends the classical local Brezis-Nirenberg problem to a nonlocal setting via a Hardy-Littlewood-Sobolev (HLS) convolution term. The problem is given by \(\begin{aligned} \left\{ \begin{array}{ll} -\Delta u=\displaystyle \Big (\int _{\Omega }\frac{u^{2_\alpha ^*}(y)}{|x-y|^\alpha }dy\Big )|u|^{2_\alpha ^*-2}u+\lambda u, \ \ & \text{ in }\ \Omega ,\\ u>0, \ \ & \text{ in }\ \Omega ,\\ u=0, \ \ & \text{ on }\ \partial \Omega , \end{array} \right. \end{aligned}\) where \(2_\alpha ^*=\frac{2N-\alpha }{N-2}=6-\alpha \) (the upper critical exponent in the sense of the HLS inequality for \(N=3\) ), \(\alpha \in (0,3)\) , and \(\lambda >0\) is a parameter. By using the Lyapunov-Schmidt reduction, we establish the existence of multiple blowing-up solutions that concentrate simultaneously around k distict points of \(\Omega \) as \(\lambda \) approaches a special value \(\lambda _0\) , which is characterized by the Green function of \(-\Delta -\lambda \) . We further extend this result to the corresponding Neumann boundary value problem. Our work fills the gap in the study of multiple blowing-up solutions for critical nonlocal Choquard equations, complementing existing results on single-bubble solutions and extending the determinant-based multi-bubble construction from local Brezis-Nirenberg problems to nonlocal settings.
