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

@article{Luo2017UniquenessOP,
  title={Uniqueness of positive solutions with concentration for the Schr{\"o}dinger–Newton problem},
  author={Peng Luo and Shuangjie Peng and Chunhua Wang},
  journal={Calculus of Variations and Partial Differential Equations},
  year={2017},
  volume={59},
  pages={1-41}
}
We are concerned with the following Schrödinger–Newton problem $$\begin{aligned} -\varepsilon ^2\Delta u+V(x)u=\frac{1}{8\pi \varepsilon ^2} \left( \int _{\mathbb {R}^3}\frac{u^2(\xi )}{|x-\xi |}d\xi \right) u,~x\in {\mathbb {R}}^3. \end{aligned}$$ - ε 2 Δ u + V ( x ) u = 1 8 π ε 2 ∫ R 3 u 2 ( ξ ) | x - ξ | d ξ u , x ∈ R 3 . For $$\varepsilon $$ ε small enough, we show the uniqueness of positive solutions concentrating at the nondegenerate critical points of V ( x ). The main tools are a local… 

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