Scaling properties of functionals and existence of constrained minimizers

• Published in: Journal of functional analysis 2011-11, Vol.261 (9), p.2486-2507
• Main Authors: Bellazzini, Jacopo, Siciliano, Gaetano
• Format: Article
• Language: English
• Subjects: Constrained minimization; Schrödinger–Poisson equations; Standing waves; Subadditivity inequality
• ISSN: 0022-1236; 1096-0783
• DOI: 10.1016/j.jfa.2011.06.014

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 R 3 i ψ t + Δ ψ − ( | x | − 1 ⁎ | ψ | 2 ) ψ + | ψ | p − 2 ψ = 0 when 2 < p < 3 . In particular we prove that I achieves its minimum on the constraint { u ∈ H 1 ( R 3 ) : ‖ u ‖ 2 = ρ } for every sufficiently small ρ > 0 . In this way we recover the case studied in Sanchez and Soler (2004) [20] for p = 8 / 3 and we complete the case studied by the authors for 3 < p < 10 / 3 in Bellazzini and Siciliano (2011) [4].