High energy positive solutions for a coupled Hartree system with Hardy-Littlewood-Sobolev critical exponents

We study the coupled Hartree system $$ left{begin{array}{ll} -Delta u+ V_1(x)u =alpha_1big(|x|^{-4}ast u^{2}big)u+beta big(|x|^{-4}ast v^{2}big)u &mbox{in} mathbb{R}^N,[1mm] -Delta v+ V_2(x)v =alpha_2big(|x|^{-4}ast v^{2}big)v +betabig(|x|^{-4}ast u^{2}big)v &mbox{in} mathbb{R}^N, end{array}right. $$ where $Ngeq 5$, $beta>max{alpha_1,alpha_2}geqmin{alpha_1,alpha_2}>0$, and $V_1,,V_2in L^{N/2}(mathbb{R}^N)cap L_{text{loc}}^{infty}(mathbb{R}^N)$ are nonnegative potentials. This system is critical in the sense of the Hardy-Littlewood-Sobolev inequality. For the system with $V_1=V_2=0$ we employ moving sphere arguments in integral form to classify positive solutions and to prove the uniqueness of positive solutions up to translation and dilation, which is of independent interest. Then using the uniqueness property, we establish a nonlocal version of the global compactness lemma and prove the existence of a high energy positive solution for the system assuming that $|V_1|_{L^{N/2}(mathbb{R}^N)}+|V_2|_{L^{N/2}(mathbb{R}^N)}>0$ is suitably small.