A coupled Hartree system with Hardy-Littlewood-Sobolev critical exponent: existence and multiplicity of high energy positive solutions

This paper deals with a coupled Hartree system with Hardy-Littlewood-Sobolev critical exponent \begin{equation*} \begin{cases} -Δu+(V_1(x)+λ_1)u=μ_1(|x|^{-4}*u^{2})u+β(|x|^{-4}*v^{2})u, \ \ &x\in R^N, -Δv+(V_2(x)+λ_2)v=μ_2(|x|^{-4}*v^{2})v+β(|x|^{-4}u^{2})v, \ \ &x\in R^N, \end{cases} \end{equation} where $N\ge 5$, ${\lambda }_1$, ${\lambda }_2\ge 0$ with ${\lambda }_1+{\lambda }_2\mathrm{\ne }0$, $V_1(x),V_2(x)\in L^{\frac{N}{2}}(R^N)$ are nonnegative functions and ${\mu }_1$, ${\mu }_2$, $\beta$ are positive constants. Such system arises from mathematical models in Bose-Einstein condensates theory and nonlinear optics. By variational methods combined with degree theory, we prove some results about the existence and multiplicity of high energy positive solutions under the hypothesis $\beta >\max \{{\mu }_1,{\mu }_2\}$