Stable standing waves for a NLS on star graphs as local minimizers of the constrained energy

被引:51
作者
Adami, Riccardo [1 ]
Cacciapuoti, Claudio [2 ]
Finco, Domenico [3 ]
Noja, Diego [4 ]
机构
[1] Politecn Torino, Dipartimento Sci Matemat GL Lagrange, Corso Duca Abruzzi 24, I-10129 Turin, Italy
[2] Univ Insubria, Dipartimento Sci & Alta Tecnol, Via Valleggio 11, I-22100 Como, Italy
[3] Univ Telemat Int Uninettuno, Fac Ingn, Corso Vittorio Emanuele 2,39, I-00186 Rome, Italy
[4] Univ Milano Bicocca, Dipartimento Matemat & Applicaz, Via R Cozzi 53, I-20125 Milan, Italy
关键词
NONLINEAR SCHRODINGER-EQUATION; ORBITAL STABILITY; STATES; MINIMIZATION;
D O I
10.1016/j.jde.2016.01.029
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
On a star graph made of N >= 3 halflines (edges) we consider a Schrodinger equation with a subcritical power-type nonlinearity and an attractive delta interaction located at the vertex. From previous works it is known that there exists a family of standing waves, symmetric with respect to the exchange of edges, that can be parametrized by the mass (or L-2-norm) of its elements. Furthermore, if the mass is small enough, then the corresponding symmetric standing wave is a ground state and, consequently, it is orbitally stable. On the other hand, if the mass is above a threshold value, then the system has no ground state. Here we prove that orbital stability holds for every value of the mass, even if the corresponding symmetric standing wave is not a ground state, since it is anyway a local minimizer of the energy among functions with the same mass. The proof is based on a new technique that allows to restrict the analysis to functions made of pieces of soliton, reducing the problem to a finite-dimensional one. In such a way, we do not need to use direct methods of Calculus of Variations, nor linearization procedures. (C) 2016 Elsevier Inc. All rights reserved.
引用
收藏
页码:7397 / 7415
页数:19
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