This paper deals with the following equation \(\begin{aligned} -\Delta u =K(|x'|, x'')\Big (|x|^{-\alpha }*(K(|x'|, x'')|u|^{2^{*}_{\alpha }})\Big )|u|^{2^{*}_{\alpha }-2}u\hspace{4.14mm}\text{ in }\hspace{1.14mm} {\mathbb {R}}^N, \end{aligned}\) where \(N\ge 5\) , \(5-\frac{6}{N-2}<\alpha \le 4\) , \(2^{*}_{\alpha }=\frac{2N-\alpha }{N-2}\) is the so-called upper critical exponent in the Hardy-Littlewood-Sobolev inequality and \(K(|x'|, x'')\) , where \((x',x'')\in {\mathbb {R}}^2\times {\mathbb {R}}^{N-2}\) , is bounded and nonnegative. Under proper assumptions on the potential function K, we obtain the existence of infinitely many solutions for the nonlocal critical equation by using a finite dimensional reduction argument and local Pohožaev identities. It is a remarkable fact that the order of the Riesz potential influences the existence/non-existence of solutions.
