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Compactness properties of critical nonlinearities and nonlinear Schrödinger equations

Part of: Equations of mathematical physics and other areas of application Partial differential equations

Published online by Cambridge University Press:  05 April 2013

David G. Costa ,
João Marcos do Ó  and
K. Tintarev
David G. Costa
Affiliation:
Department of Mathematical Sciences, University of Nevada Las Vegas, Box 454020, Las Vegas, NV 89154-4020, USA (costa@unlv.nevada.edu)
João Marcos do Ó
Affiliation:
Department of Mathematics, Universidade Federal de Paraiba, 58051-900 João Pessoa, PB, Brazil (jmbo@pq.cnpq.br)
K. Tintarev
Affiliation:
Department of Mathematics, Uppsala University, PO Box 480, 75106 Uppsala, Sweden (kyril.tintarev@math.uu.se)
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Abstract

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We prove the compactness of critical Sobolev embeddings with applications to nonlinear singular Schrödinger equations and provide a unified treatment in dimensions N > 2 and N = 2, based on a somewhat unexpectedly broad array of parallel properties between spaces and H10 of the unit disc. These properties include Leray inequality for N = 2 as a counterpart of Hardy inequality for N > 2, pointwise estimates by ground states r(2−N)/2 and of the respective Hardy-type inequalities, as well as compactness of the limiting Sobolev embeddings once the Sobolev norm is appended by a potential term whose growth at singularities exceeds that of the corresponding Hardy-type potential.

Keywords

Trudinger–Moser inequalityHardy inequalitycritical nonlinearitySchrödinger equationsingular potentials

MSC classification

Secondary: 35B33: Critical exponents 35B38: Critical points 35Q55: NLS-like equations (nonlinear Schrödinger)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 56 , Issue 2 , June 2013 , pp. 427 - 441
Copyright
Copyright © Edinburgh Mathematical Society 2013

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