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Compactness properties of critical nonlinearities and nonlinear Schrödinger equations

Published online by Cambridge University Press:  05 April 2013

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.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 2013

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