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Nonlinear Schrödinger equations with Hardy potential and critical nonlinearities

Author: Didier Smets
Journal: Trans. Amer. Math. Soc. 357 (2005), 2909-2938
MSC (2000): Primary 35J10, 35J70, 58J37
Published electronically: December 29, 2004
MathSciNet review: 2139932
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Abstract: We study a time-independent nonlinear Schrödinger equation with an attractive inverse square potential and a nonautonomous nonlinearity whose power is the critical Sobolev exponent. The problem shares a strong resemblance with the prescribed scalar curvature problem on the standard sphere. Particular attention is paid to the blow-up possibilities, i.e. the critical points at infinity of the corresponding variational problem. Due to the strong singularity in the potential, some new phenomenon appear. A complete existence result is obtained in dimension 4 using a detailed analysis of the gradient flow lines.

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Additional Information

Didier Smets
Affiliation: Laboratoire J.L. Lions, Université Pierre & Marie Curie, 175 rue du chevaleret, 75013 Paris, France

Keywords: Hardy potential, critical Sobolev exponent, prescribed scalar curvature
Received by editor(s): February 11, 2002
Received by editor(s) in revised form: January 13, 2004
Published electronically: December 29, 2004
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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