Existence and orbital stability of the ground states with prescribed mass for the L2-critical and supercritical NLS on bounded domains

where $B_{1} \subset ℝ^{N}$ 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 $L^{2}$ -critical and subcritical, i.e., $1 < p \leq 1 + 4 ∕ N$ , while they are stable for almost every $ρ$ in the $L^{2}$ -supercritical regime $1 + 4 ∕ N < p < 2^{*} - 1$ . The proofs are obtained in connection with the study of a variational problem with two constraints of independent interest: to maximize the $L^{p + 1}$ -norm among functions having prescribed $L^{2}$ - and $H_{0}^{1}$ -norms.

Keywords

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

Mathematical Subject Classification 2010

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

Milestones

Received: 24 July 2013

Revised: 29 September 2014

Accepted: 2 November 2014

Published: 5 February 2015

Authors

Benedetta Noris
INdAM-COFUND Marie Curie Fellow Laboratoire de Mathématiques Université de Versailles Saint-Quentin-en-Yvelines 45 avenue des Étas-Unis 78035 Versailles France

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 Portugal

Gianmaria Verzini
Dipartimento di Matematica Politecnico di Milano Piazza Leonardo da Vinci 32 20133 Milano Italy

Vol. 7, No. 8, 2014

Benedetta Noris, Hugo Tavares and Gianmaria Verzini

Abstract

Keywords

Mathematical Subject Classification 2010

Milestones

Authors