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

Abstract: 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.