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, $$-\mathrm{\Delta }u+V(\mathrm{\mid }{x}^{\mathrm{\prime }}\mathrm{\mid },x^{\mathrm{\prime }\mathrm{\prime }})u=(\mathrm{\mid }x{\mathrm{\mid }}^{-4}\ast \mathrm{\mid }u{\mathrm{\mid }}^2)uin{\mathbb{R}}^6,$$ where $(x^{\mathrm{\prime }},x^{\mathrm{\prime }\mathrm{\prime }})\in {\mathbb{R}}^2\times {\mathbb{R}}^4$, $V(\mathrm{\mid }{x}^{\mathrm{\prime }}\mathrm{\mid },x^{\mathrm{\prime }\mathrm{\prime }})$ is a bounded nonnegative function in ${\mathbb{R}}^{+}\times {\mathbb{R}}^4$, and $\ast$ 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 $r^{2}V(r,x^{\mathrm{\prime }\mathrm{\prime }})$ has a topologically nontrivial critical point then the problem admits infinitely many solutions with arbitrary large energies.