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Nonlinear Schrödinger equations with strongly singular potentials

Published online by Cambridge University Press:  04 August 2010

Jacopo Bellazzini
Affiliation:
Dipartimento di Matematica Applicata ‘U. Dini’, Università di Pisa, via Buonarroti 1/c, 56127 Pisa, Italy (j.bellazzini@ing.unipi.it; bonanno@mail.dm.unipi.it)
Claudio Bonanno
Affiliation:
Dipartimento di Matematica Applicata ‘U. Dini’, Università di Pisa, via Buonarroti 1/c, 56127 Pisa, Italy (j.bellazzini@ing.unipi.it; bonanno@mail.dm.unipi.it)

Abstract

We look for standing waves for nonlinear Schrödinger equations

with cylindrically symmetric potentials g vanishing at infinity and non-increasing, and a C1 nonlinear term satisfying weak assumptions. In particular, we show the existence of standing waves with non-vanishing angular momentum with prescribed L2 norm. The solutions are obtained via a minimization argument, and the proof is given for an abstract functional which presents a lack of compactness. As a specific case, we prove the existence of standing waves with non-vanishing angular momentum for the nonlinear hydrogen atom equation.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2010

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