Uniqueness of positive solutions with concentration for the Schrödinger–Newton problem

We are concerned with the following Schrodinger–Newton problem $$\begin{aligned} -\varepsilon ^2\Delta u+V(x)u=\frac{1}{8\pi \varepsilon ^2} \left( \int _{\mathbb {R}^3}\frac{u^2(\xi )}{|x-\xi |}d\xi \right) u,~x\in {\mathbb {R}}^3. \end{aligned}$$For $$\varepsilon $$ 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.