Existence and qualitative properties of solutions for a Choquard-type equation with Hardy potential

In this paper, we study the existence and qualitative properties of positive solutions to a Choquard-type equation with Hardy potential. We develop a nonlocal version of concentration-compactness principle involving the Hardy potential to study the existence and the asymptotic behavior of positive solutions by transforming the original problem into a new nonlocal problem in the weighted Sobolev space. Moreover, we obtain the symmetry of solutions by using the moving plane method.