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Scaling properties of functionals and existence of constrained minimizers

Bellazzini, Jacopo and Siciliano, Gaetano (2011) Scaling properties of functionals and existence of constrained minimizers. Journal of Functional Analysis, Vol. 261 (9), p. 2486-2507. ISSN 0022-1236. eISSN 1096-0783. Article.

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DOI: 10.1016/j.jfa.2011.06.014

Abstract

In this paper we develop a new method to prove the existence of minimizers for a class of constrained minimization problems on Hilbert spaces that are invariant under translations. Our method permits to exclude the dichotomy of the minimizing sequences for a large class of functionals. We introduce family of maps, called scaling paths, that permits to show the strong subadditivity inequality. As byproduct the strong convergence of the minimizing sequences (up to translations) is proved. We give an application to the energy functional I associated to the Schrödinger–Poisson equation in R3

iψt + Δψ − (|x|−1 * |ψ|2)ψ + |ψ|p−2ψ = 0
when 2 < p < 3. In particular we prove that I achieves its minimum on the constraint {u ∈ H1(R3): ||u||2 = ρ} for every sufficiently small ρ >0. In this way we recover the case studied in Sanchez and Soler (2004) for p = 8/3 and we complete the case studied by the authors for 3 < p <10/3 in Bellazzini and Siciliano (2011).

Item Type:Article
ID Code:6226
Status:Published
Refereed:Yes
Uncontrolled Keywords:Constrained minimization, subadditivity inequality, Schrödinger–Poisson equations, standing waves
Subjects:Area 01 - Scienze matematiche e informatiche > MAT/05 Analisi matematica
Divisions:001 Università di Sassari > 01 Dipartimenti > Scienze botaniche, ecologiche e geologiche
Publisher:Academic Press / Elsevier
ISSN:0022-1236
eISSN:1096-0783
Copyright Holders:© 2011 Elsevier
Deposited On:07 Jul 2011 11:35

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