Abstract

We are concerned with the following Schrödinger–Newton problem $$\begin{aligned} -\varepsilon ^2\Delta u+V(x)u=\frac{1}{8\pi \varepsilon ^2} \left( \int _{\mathbb {R}^3}\frac{u^2(\xi )}{|x-\xi |}d\xi \right) u,~x\in {\mathbb {R}}^3. \end{aligned}$$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 Schrödinger–Newton problem is quite different from those of Schrödinger equations, which is mainly caused by the nonlocal term.

中文翻译：

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Schrödinger–Newton问题正解集中的唯一性

摘要

我们关注以下Schrödinger–Newton问题$$ \ begin {aligned}-\ varepsilon ^ 2 \ Delta u + V（x）u = \ frac {1} {8 \ pi \ varepsilon ^ 2} \ left（\ int _ {\ mathbb {R} ^ 3} \ frac {u ^ 2（\ xi）} {| x- \ xi |} d \ xi \ right）u，〜x \ in {\ mathbb {R}} ^ 3。\ end {aligned} $$对于\（\ varepsilon \）足够小，我们展示了集中在V（x）的简并临界点上的正解的唯一性。主要工具是本地Pohozaev类型的标识，爆炸分析和最大化原理。我们的研究结果还表明，Schrödinger-Newton问题的集中点的渐近行为与Schrödinger方程非常不同，这主要是由非局部项引起的。