Uniqueness of positive solutions with concentration for the Schrodinger-Newton problem
Uniqueness of positive solutions with concentration for the Schrodinger-Newton problem
复制标题
薛定谔-牛顿问题集中正解的唯一性
DOI：
10.1007/s00526-020-1726-6
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发表时间：
2020
影响因子：
2.1
通讯作者：
Wang Chunhua
中科院分区：
文献类型：
--
作者：
Luo Peng;Peng Shuangjie;Wang Chunhua
我们关注下面的薛定谔-牛顿问题$$\开始{对齐} -\varepidu^2\Delta u+V（x）u=\frac{1}{8\pi \varepidu^2} \left（\int _{\mathbb {R}^3}\frac{u^2（\xi）}{|x-\xi|}d\xi \right）u，~x\in {\mathbb {R}}^3.当V（x）足够小时，证明了正解集中在非退化临界点的唯一性.主要的工具是当地Pohozaev类型的身份，爆破分析和最大值原理。我们的结果还表明，薛定谔-牛顿问题的集中点的渐近行为与薛定谔方程的集中点的渐近行为有很大的不同，这主要是由非局部项引起的。
We are concerned with the following Schrödinger–Newton problem $$\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}$$Forsmall enough, we show the uniqueness of positive solutions concentrating at the nondegenerate critical points ofV(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.