Uniqueness of positive solutions with Concentration for the SchrödingerNewton problem
Abstract
We are concerned with the following SchrödingerNewton problem \begin{equation} \varepsilon^2\Delta u+V(x)u=\frac{1}{8\pi \varepsilon^2} \big(\int_{\mathbb R^3}\frac{u^2(\xi)}{x\xi}d\xi\big)u,~x\in \mathbb R^3. \end{equation} 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, blowup analysis and the maximum principle. Our results also show that the asymptotic behavior of concentrated points to SchrödingerNewton problem is quite different from those of Schrödinger equations.
 Publication:

arXiv eprints
 Pub Date:
 March 2017
 arXiv:
 arXiv:1703.00777
 Bibcode:
 2017arXiv170300777L
 Keywords:

 Mathematics  Analysis of PDEs