Constrained energy minimization  and  ground states for NLS with point defects

Riccardo Adami; Diego Noja; Nicola Visciglia

doi:10.3934/dcdsb.2013.18.1155

Article Contents

2013, Volume 18, Issue 5: 1155-1188. Doi: 10.3934/dcdsb.2013.18.1155

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Constrained energy minimization and ground states for NLS with point defects

1.
Dipartimento di Scienze Matematiche, Politecnico di Torino, C.so Duca degli Abruzzi 24, 10129 Torino
2.
Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, via R. Cozzi 53, 20125 Milano
3.
Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo, 56100 Pisa

Received: July 2012

Revised: January 2013

Published: July 2013

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Abstract

We investigate the ground states of the one-dimensional nonlinear Schrödinger equation with a defect located at a fixed point. The nonlinearity is focusing and consists of a subcritical power. The notion of ground state can be defined in several (often non-equivalent) ways. We define a ground state as a minimizer of the energy functional among the functions endowed with the same mass. This is the physically meaningful definition in the main fields of application of NLS. In this context we prove an abstract theorem that revisits the concentration-compactness method and which is suitable to treat NLS with inhomogeneities. Then we apply it to three models, describing three different kinds of defect: delta potential, delta prime interaction, and dipole. In the three cases we explicitly compute ground states and we show their orbital stability.

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Mathematics Subject Classification: Primary: 35Q55; Secondary: 35J10.

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