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

  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},
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… 

Multi-peak Positive Solutions of a Nonlinear Schrödinger–Newton Type System

Abstract In this paper, we study the following nonlinear Schrödinger–Newton type system: { - ϵ 2 ⁢ Δ ⁢ u + u - Φ ⁢ ( x ) ⁢ u = Q ⁢ ( x ) ⁢ | u | ⁢ u , x ∈ ℝ 3 , - ϵ 2 ⁢ Δ ⁢ Φ = u 2 , x ∈ ℝ 3 ,

Properties of the minimizers for a constrained minimization problem arising in Kirchhoff equation

Let $a>0,b>0$ and $V(x)\geq0$ be a coercive function in $\mathbb R^2$. We study the following constrained minimization problem on a suitable weighted Sobolev space $\mathcal{H}$: \begin{equation*}

Positive multi-peak solutions for a logarithmic Schrodinger equation

In this manuscript, we consider the logarithmic Schrodinger equation \begin{eqnarray*} -\varepsilon^2\Delta u+V(x)u=u\log u^{2},\,\,\,u>0, & \text{in}\,\,\,\mathbb{R}^{N}, \end{eqnarray*} where

Local uniqueness and the number of concentrated solutions for nonlinear Schrödinger equations with non-admissible potential

We revisit the following nonlinear Schrödinger equation }0,u\in {H}^{1}\left({\mathbb{R}}^{N}\right),$?> −ε2Δu+V(x)u=up−1,u>0,u∈H1(RN), where ɛ > 0 is a small parameter, N ⩾ 2, 2 < p < 2*. Here we

Construction of infinitely many solutions for a critical Choquard equation via local Pohožaev identities

In this paper, we study a class of critical Choquard equations with axisymmetric potentials, -Δu+V(|x′|,x′′)u=(|x|-4∗|u|2)uinR6,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym}

Uniqueness of Single Peak Solutions for Coupled Nonlinear Gross-Pitaevskii Equations with Potentials

For a couple of singularly perturbed Gross-Pitaevskii equations, we first prove that the single peak solutions, if they concentrate on the same point, are unique provided that the Taylor’s expansion

Existence and local uniqueness of normalized peak solutions for a Schrödinger-Newton system

In this paper, we investigate the existence and local uniqueness of normalized peak solutions for a Schrodinger-Newton system under the assumption that the trapping potential is degenerate and has

New type of solutions for the nonlinear Schr\"odinger-Newton system

is a nonlinear system obtained by coupling the linear Schrödinger equation of quantum mechanics with the gravitation law of Newtonian mechanics. Wei and Yan in (Calc. Var. Partial Differential

Infinitely many solutions for Schr\"{o}dinger-Newton equations

Here V is a given external potential and Ψ is the Newtonian gravitational potential. The latter model was proposed in [22], where the wave function u represents a stationary solution for a quantum



Uniqueness of positive bound states with multi-bump for nonlinear Schrödinger equations

We are concerned with the following nonlinear Schrödinger equation $$\begin{aligned} -\varepsilon ^2\Delta u+ V(x)u=|u|^{p-2}u,\quad u\in H^1(\mathbb {R}^N),


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

Uniqueness of positive multi-lump bound states of nonlinear Schrödinger equations

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

On concentration of positive bound states of nonlinear Schrödinger equations

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

Strongly interacting bumps for the Schrödinger–Newton equations

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

Uniqueness and nondegeneracy of ground states for Choquard equations in three dimensions

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,

Existence of Bound States for Schrödinger-Newton Type Systems

Abstract In this paper we consider the following elliptic system in ℝ3 where K(x), α(x) are non-negative real functions defined on ℝ3 so that . When K(x) ≡ K∞ and α(x) ≡ α∞ we have already proved the

On a nonlinear elliptic equation involving the critical sobolev exponent: The effect of the topology of the domain

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