Normalized ground states for the NLS equation with combined nonlinearities: The Sobolev critical case

2020/01/01 by Nicola Soave · 53 citations
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Context (archaeology) #Critical mass (sociodynamics) #Instability #Lambda #Mathematical analysis #Mathematical physics #Mathematics #Nonlinear Schrödinger equation #Nonlinear system #Perturbation (astronomy) #Physics #Quantum mechanics #Schrödinger equation #Sobolev space #Space (punctuation) #Spectral Theory in Mathematical Physics #Stability and Controllability of Differential Equations #Supercritical fluid

paper · pdf · doi:10.1016/j.jfa.2020.108610

published in Journal of Functional Analysis 279(6), 108610 (Elsevier BV)

openalex publication_date 2020/05/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We study existence and properties of ground states for the nonlinear Schrödinger equation with combined power nonlinearities −Δu=λu+μ|u|q−2u+|u|2javax.xml.bind.JAXBElement@4d419c48−2uin RN, N≥3, having prescribed mass ∫RN|u|2=a2, in the Sobolev critical case. For a L2-subcritical, L2-critical, of L2-supercritical perturbation μ|u|q−2u we prove several existence/non-existence and stability/instability results. This study can be considered as a counterpart of the Brezis-Nirenberg problem in the context of normalized solutions, and seems to be the first contribution regarding existence of normalized ground states for the Sobolev critical NLSE in the whole space RN.

Citations

• Nonlinear Schr�dinger equations and sharp interpolation estimates

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