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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Bibliographic Details
Published inJournal of functional analysis Vol. 280; no. 11; p. 108989
Main Authors Bieganowski, Bartosz, Mederski, Jarosław
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.06.2021
Subjects
Nonlinear scalar field equation
Nonlinear Schrödinger equations
Normalized ground states
Pohozaev manifold
35J20
Normalized ground states
35J60
35Q55
Nonlinear scalar field equation
Nonlinear Schrödinger equations
Pohozaev manifold
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Summary: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.
ISSN:0022-1236
1096-0783
DOI:10.1016/j.jfa.2021.108989