In this paper, we study the following fractional Choquard problem: \( {\left\{ \begin{array}{ll} (-\varDelta )^s u = \lambda \vert u\vert ^{q-2} u + g \left( I_\mu * \bigl (g \vert u\vert ^{2_{\mu ,s}^*}\bigr ) \right) \vert u\vert ^{2_{\mu ,s}^* - 2} u,& \text {in } \varOmega , \\ u = 0, & \text {on } \mathbb {R}^N \setminus \varOmega , \end{array}\right. } \) where \(\varOmega \subset \mathbb {R}^N\) ( \(N \ge 2s\) ) is a bounded domain with continuous boundary, \(0< \mu < N\) , and \(I_\mu \) is defined for \(x \in \mathbb {R}^N \setminus \{0\}\) by \(I_\mu (x) = \frac{1}{\vert x\vert ^\mu }\) . Here, \( 2_{\mu ,s}^* = \frac{2N - \mu }{N - 2s} \quad \text {and} \quad q \in [2, 2_s^*), \text { with } 2_s^* = \frac{2N}{N - 2s}. \) By employing variational methods and the Nehari manifold technique, we establish the existence of multiple positive solutions. In particular, we show that the number of such solutions is closely related to the set of global maxima of the coefficient function \( g \) appearing in the nonlocal critical nonlinearity provided that \(q=2\) with \(N\ge 4s\) , or \(q\in (2,2_s^*)\) with \(N > \tfrac{2s(q+2)}{q}\) . Moreover, we prove that every weak solution obtained is bounded and Hölder continuous.
