A Choquard type equation involving mixed local and nonlocal operators

In the present work, we are concerned with an elliptic problem involving a mixed order diffusion operator with both local and nonlocal parts and a critical nonlinearity of Hartree type. We first investigate the corresponding Hardy-Littlewood-Sobolev inequality and establish the optimal constant. Using variational methods and a Pohožaev identity we then show existence and nonexistence results for the corresponding subcritical perturbation problem.