Coron problem for nonlocal equations invloving Choquard nonlinearity

Abstract

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

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