  The paper is concerned with the existence and multiplicity of positive
solutions of the nonhomogeneous Choquard equation over an annular type bounded
domain. Precisely, we consider the following equation
  \[
  -\De u =
\left(\int_{\Om}\frac{|u(y)|^{2^*_{\mu}}}{|x-y|^{\mu}}dy\right)|u|^{2^*_{\mu}-2}u+f
\; \text{in}\;
  \Om,\quad
  u = 0 \; \text{ on } \pa \Om ,
  \]
  where $\Om$ is a smooth bounded annular domain in $\mathbb{R}^N( N\geq 3)$,
$2^*_{\mu}=\frac{2N-\mu}{N-2}$, $f \in L^{\infty}(\Om)$ and $f \geq 0$. We
prove the existence of four positive solutions of the above problem using the
Lusternik-Schnirelmann theory and varitaional methods, when the inner hole of
the annulus is sufficiently small.
