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.