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In this article, we consider the problem $$ -\Delta u =\Big(\int_{\mathbb{R}^{N}} \frac{|u|^{2^{*}_{\mu}}}{|x-y|^{\mu}}\,\mathrm{d}y \Big) |u|^{2^{*}_{\mu}-2}u + f(x,u) \quad\text{in }\mathbb{R}^{N}, $$ where $N\geqslant3$, $\mu\in(0,N)$ and $2^{*}_{\mu}=\frac{2N-\mu}{N-2}$. Under suitable assumptions on $f(x,u)$, we establish the existence and non-existence of nontrivial solutions via the variational method.