Existence of prescribed \(L^2\)-norm solutions for a class of Schrödinger-Poisson equation
type: Item
instance of: scholarly article
title: Existence of prescribed \(L^2\)-norm solutions for a class of Schrödinger-Poisson equation
zbMATH Open document ID: 1298.35193
DOI: 10.1155/2013/398164
author: Zeng Liu
author: Yuanze Wu
author: Yisheng Huang
published in: Abstract and Applied Analysis
publication date: 2014-06-23T00:00:00Z
review text: Summary: By using the standard scaling arguments, we show that the infimum of the following minimization problem: \[ I_{\rho^2}= \text{inf}\Biggl\{(1/2) \int_{\mathbb{R}^3} |\nabla u|^2 dx+ (1/4) \iint_{\mathbb{R}^3} (|u(x)|^2|u(y)|^2/|x- y|)\,dx\,dy-(1/p) \int_{\mathbb{R}^3} |u|^p dx: u\in B_\rho\Biggr\} \] can be achieved for \(p\in (2,3)\) and \(\rho>0\) small, where \(B_\rho:=\{u\in H^1(\mathbb{R}^3):\| u\|_2= \rho\}\). Moreover, the properties of \(I_{\rho^2}/\rho^2\) and the associated Lagrange multiplier \(\lambda_\rho\) are also given if \(p\in (2,8/3]\).
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 35J20
zbMATH DE Number: 6306797
zbMATH Keywords: minimization problem
zbMATH Keywords: nonlinear Schrödinger-Poisson type equation
Wikidata QID: Q58916166
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2013/398164
OpenAlex ID: W1993505198
cites work: LOCAL APPROXIMATION FOR THE HARTREE–FOCK EXCHANGE POTENTIAL: A DEFORMATION APPROACH
cites work: On an Exchange Interaction Model for Quantum Transport: The Schrödinger–Poisson–Slater System
cites work: Q4434430
cites work: Long-time dynamics of the Schrödinger-Poisson-Slater system
cites work: The Schrödinger-Poisson equation under the effect of a nonlinear local term
cites work: On Schrödinger-Poisson systems
cites work: On the Schrödinger-Maxwell equations under the effect of a general nonlinear term
cites work: Positive solutions for some non-autonomous Schrödinger-Poisson systems
cites work: Q4826007
cites work: Solitary waves for nonlinear Klein–Gordon–Maxwell and Schrödinger–Maxwell equations
cites work: Schrödinger-Poisson system with steep potential well
cites work: On the existence of solutions for the Schrödinger-Poisson equations
cites work: Sharp nonexistence results of prescribed \(L^2\)-norm solutions for some class of Schrödinger-Poisson and quasi-linear equations
cites work: Existence and instability of standing waves with prescribed norm for a class of Schrödinger-Poisson equations
cites work: The concentration-compactness principle in the calculus of variations. The locally compact case. I
cites work: The concentration-compactness principle in the calculus of variations. The locally compact case. II
cites work: Stable standing waves for a class of nonlinear Schrödinger-Poisson equations
cites work: Scaling properties of functionals and existence of constrained minimizers
cites work: Hylomorphic solitons on lattices
cites work: Existence and Stability of Standing Waves For Schrödinger-Poisson-Slater Equation
cites work: On the radiality of constrained minimizers to the Schrödinger-Poisson-Slater energy
description: scientific article; zbMATH DE number 6306797