Normalized ground states of the nonlinear Schrödinger equation with at least mass critical growth

We propose a simple minimization method to show the existence of least energy solutions to the normalized problem{−Δu+λu=g(u)inRN,N≥3,u∈H1(RN),∫RN|u|2dx=ρ>0, where ρ is prescribed and (λ,u)∈R×H1(RN) is to be determined. The new approach based on the direct minimization of the energy functional on...

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Published in:Journal of functional analysis 2021-06, Vol.280 (11), p.108989, Article 108989
Main Authors: Bieganowski, Bartosz, Mederski, Jarosław
Format: Article
Language:English
Subjects:
Nonlinear scalar field equation
Nonlinear Schrödinger equations
Normalized ground states
Pohozaev manifold
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description We propose a simple minimization method to show the existence of least energy solutions to the normalized problem{−Δu+λu=g(u)inRN,N≥3,u∈H1(RN),∫RN|u|2dx=ρ>0, where ρ is prescribed and (λ,u)∈R×H1(RN) is to be determined. The new approach based on the direct minimization of the energy functional on the linear combination of Nehari and Pohozaev constraints intersected with the closed ball in L2(RN) of radius ρ is demonstrated, which allows to provide general growth assumptions imposed on g. We cover the most known physical examples and nonlinearities with growth considered in the literature so far as well as we admit the mass critical growth at 0.
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subjects Nonlinear scalar field equation
Nonlinear Schrödinger equations
Normalized ground states
Pohozaev manifold
title Normalized ground states of the nonlinear Schrödinger equation with at least mass critical growth
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