Nonlinear Schrödinger equations with Hardy potential and critical nonlinearities

Smets, Didier

doi:10.1090/S0002-9947-04-03769-9

Nonlinear Schrödinger equations with Hardy potential and critical nonlinearities
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by Didier Smets

Trans. Amer. Math. Soc. 357 (2005), 2909-2938

DOI: https://doi.org/10.1090/S0002-9947-04-03769-9

Published electronically: December 29, 2004

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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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Nonlinear Schrödinger equations with Hardy potential and critical nonlinearities
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Abstract: