# 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}
}
• 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…
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We consider the following nonlinear Schrodinger–Poisson system in ℝ3 $$\left\{ \begin{array}{@{}l@{\quad}l} -\Delta u+u+K(|y|)\Phi(y)u=Q(|y|)|u|^{p-1}u, & y\in {\mathbb R}^3,\\[3pt] -\Delta • Mathematics • 2003 Abstract. In this paper we are concerned with multi-lump bound states of the nonlinear Schrödinger equation  i\hbar \frac{\partial \psi}{\partial t} \: = \: -\hbar^2 \Delta \psi + V\psi - \gamma AbstractWe study the concentration behavior of positive bound states of the nonlinear Schrödinger equation$$ih\frac{{\partial \psi }}{{\partial t}} = \frac{{ - h^2 }}{{2m}}\Delta \psi + V\left( x
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We study concentrated bound states of the Schrodinger–Newton equations h2Δψ−E(x)ψ+Uψ=0, ψ>0, x∊R3; ΔU+12|ψ|2=0, x∊R3; ψ(x)→0, U(x)→0 as |x|→∞. Moroz et al. [“An analytical approach to the
We obtain uniqueness and nondegeneracy results for ground states of Choquard equations $$-\Delta u+u=\left( |x|^{-1}*|u|^{p}\right) |u|^{p-2}u$$-Δu+u=|x|-1∗|u|p|u|p-2u in $$\mathbb {R}^{3}$$R3,
• Mathematics
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Soit Ω un ensemble ouvert borne regulier et connexe de R N , N≥3. On considere u:Ω→R telle que −Δu=u (N+2)/(N−2) dans Ω, u>0 dans Ω, u=0 sur ∂Ω. On note par Hd(Ω; Z 2 ) l'homologie de diemnsion d de