  In this article, we study an elliptic problem involving an operator of mixed
order with both local and nonlocal aspects and in the presence of critical
nonlinearity of Hartree type. To this end, we first investigate the
corresponding Hardy-Littlewood-Sobolev inequality and detect the optimal
constant. Using variational methods and a Poho\v{z}aev identity we then show
the existence and nonexistence results for the corresponding subcritical
perturbation problem.
