Construction of infinitely many solutions for a critical Choquard equation via local Pohožaev identities

In this paper, we study a class of critical Choquard equations with axisymmetric potentials, $$\begin{aligned} -\Delta u+ V(|x'|,x'')u =\Big (|x|^{-4}*|u|^{2}\Big )u\text{ in } \mathbb {R}^6, \end{aligned}$$ where $$(x',x'')\in \mathbb {R}^2\times \mathbb {R}^{4}$$ , $$V(|x'|, x'')$$ is a bounded nonnegative function in $$\mathbb {R}^{+}\times \mathbb {R}^{4}$$ , and $$*$$ stands for the standard convolution. The equation is critical in the sense of the Hardy-Littlewood-Sobolev inequality. By applying a finite dimensional reduction argument and developing novel local Pohožaev identities, we prove that if the function $$r^2V(r,x'')$$ has a topologically nontrivial critical point then the problem admits infinitely many solutions with arbitrary large energies.