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2014 Existence and orbital stability of the ground states with prescribed mass for the $L^2$-critical and supercritical NLS on bounded domains
Benedetta Noris, Hugo Tavares, Gianmaria Verzini
Anal. PDE 7(8): 1807-1838 (2014). DOI: 10.2140/apde.2014.7.1807

Abstract

Given ρ>0, we study the elliptic problem

 find  ( U , λ ) H 0 1 ( B 1 ) ×  such that  Δ U + λ U = U p , B 1 U 2 d x = ρ , U > 0 ,

where B1N is the unitary ball and p is Sobolev-subcritical. Such a problem arises in the search for solitary wave solutions for nonlinear Schrödinger equations (NLS) with power nonlinearity on bounded domains. Necessary and sufficient conditions (about ρ, N and p) are provided for the existence of solutions. Moreover, we show that standing waves associated to least energy solutions are orbitally stable for every ρ (in the existence range) when p is L2-critical and subcritical, i.e., 1<p1+4N, while they are stable for almost every ρ in the L2-supercritical regime 1+4N<p<21. The proofs are obtained in connection with the study of a variational problem with two constraints of independent interest: to maximize the Lp+1-norm among functions having prescribed L2- and H01-norms.

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Benedetta Noris. Hugo Tavares. Gianmaria Verzini. "Existence and orbital stability of the ground states with prescribed mass for the $L^2$-critical and supercritical NLS on bounded domains." Anal. PDE 7 (8) 1807 - 1838, 2014. https://doi.org/10.2140/apde.2014.7.1807

Information

Received: 24 July 2013; Revised: 29 September 2014; Accepted: 2 November 2014; Published: 2014
First available in Project Euclid: 20 December 2017

zbMATH: 1314.35168
MathSciNet: MR3318740
Digital Object Identifier: 10.2140/apde.2014.7.1807

Subjects:
Primary: 35B35 , 35C08 , 35J20 , 35Q55

Keywords: Ambrosetti–Prodi-type problem , constrained critical points , Gagliardo–Nirenberg inequality , singular perturbations

Rights: Copyright © 2014 Mathematical Sciences Publishers

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