Uniqueness of positive solutions with concentration for the Schrodinger-Newton problem

We are concerned with the following Schrodinger-Newton problem -epsilon(2) Delta u + V(x)u = 1/8 pi epsilon(2)(integral(R3) u(2)(xi)/vertical bar x-xi vertical bar d xi)u, x is an element of R-3. For epsilon small enough, we show the uniqueness of positive solutions concentrating at the nondegenerate critical points of V(x). The main tools are a local Pohozaev type of identity, blow-up analysis and the maximum principle. Our results also show that the asymptotic behavior of concentrated points to Schrodinger-Newton problem is quite different from those of Schrodinger equations, which is mainly caused by the nonlocal term.