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

In this paper, we study a class of the critical Choquard equations with axisymmetric potentials, $$ -\Delta u+ V(|x'|,x'')u =\Big(|x|^{-4}\ast |u|^{2}\Big)u\hspace{4.14mm}\mbox{in}\hspace{1.14mm} \mathbb{R}^6, $$ 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.