Coron problem for nonlocal equations invloving Choquard nonlinearity

Abstract: We study the problem [ -\De u = \left(\int_{\Om}\frac{|u(y)|^{{2_{\mu}}}{|x-y|{\mu}}dy\right)|u|{2^_{\mu}-2}u,} \; \text{in}\; \Om,\quad u = 0 \; \text{ on } \pa \Om , ] where $\Om$ is a smooth bounded domain in $\mathbb{R}^N( N\geq 3)$, $2^{*_{\mu}=\frac{2N-\mu}{N-2}$.} we prove the existence of a positive solution of the above problem in an annular type domain when the inner hole is sufficiently small.