• DOI:10.1007/s00526-020-1726-6
• Corpus ID: 119326293

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

@article{Luo2020UniquenessOP,
  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={2020},
  volume={59},
  pages={1-41}
}

• Peng Luo, Shuangjie Peng, Chunhua Wang
• Published 2 March 2017
• Mathematics
• Calculus of Variations and Partial Differential Equations

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 Pohozaev type of identity, blow-up analysis and the maximum principle. Our results also show that the asymptotic behavior of concentrated points to Schrödinger–Newton problem is quite different from those of Schrödinger equations, which is mainly caused by the nonlocal term.