Existence of infinitely many solutions for a critical Hartree type equation with potential: local Poho\v{z}aev identities methods

This paper deals with the following equation $$-\Delta u =K(|x'|, x'')\Big(|x|^{-\alpha}\ast (K(|x'|, x'')|u|^{2^{\ast}_{\alpha}})\Big) |u|^{2^{\ast}_{\alpha}-2}u\quad\mbox{in}\ \mathbb{R}^N,$$ where $N\geq5$, $\alpha>5-\frac{6}{N-2}$, $2^{\ast}_{\alpha}=\frac{2N-\alpha}{N-2}$ 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 Poho\v{z}aev identities. It is a remarkable fact that the order of the Riesz potential influences the existence/non-existence of solutions.