Vol. 7, No. 8, 2014

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
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 and Gianmaria Verzini

Vol. 7 (2014), No. 8, 1807–1838

Given ρ > 0, we study the elliptic problem

 find (U,λ) H01(B1) ×  such that  ΔU + λU = Up , B1U2dx = ρ, U > 0,

where B1 N 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 < p 1 + 4N, while they are stable for almost every ρ in the L2-supercritical regime 1 + 4N < p < 2 1. 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.

Gagliardo–Nirenberg inequality, constrained critical points, Ambrosetti–Prodi-type problem, singular perturbations
Mathematical Subject Classification 2010
Primary: 35B35, 35J20, 35Q55, 35C08
Received: 24 July 2013
Revised: 29 September 2014
Accepted: 2 November 2014
Published: 5 February 2015
Benedetta Noris
INdAM-COFUND Marie Curie Fellow
Laboratoire de Mathématiques
Université de Versailles Saint-Quentin-en-Yvelines
45 avenue des Étas-Unis
78035 Versailles
Hugo Tavares
Universidade de Lisboa
Centro de Matemática e Aplicações Fundamentais and Faculdade de Ciências da Universidade de Lisboa
Avenida Professor Gama Pinto 2
1649-003 Lisboa
Gianmaria Verzini
Dipartimento di Matematica
Politecnico di Milano
Piazza Leonardo da Vinci 32
20133 Milano