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The paper is concerned with the existence and multiplicity of positivesolutions of the nonhomogeneous Choquard equation over an annular type boundeddomain. 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 $f \geq 0$. Weprove the existence of four positive solutions of the above problem using theLusternik-Schnirelmann theory and varitaional methods, when the inner hole ofthe annulus is sufficiently small.

Critical growth elliptic problems involving Hardy-Littlewood-Sobolev critical exponent in non-contractible domains