Nonlinear Schrödinger equations      with inverse-square potentials      in two dimensional space

Toshiyuki Suzuki

doi:10.3934/proc.2015.1019

Article Contents

2015, Volume 2015: 1019-1024. Doi: 10.3934/proc.2015.1019 open access

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Nonlinear Schrödinger equations with inverse-square potentials in two dimensional space

Toshiyuki Suzuki^1,

1.
Department of Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601

Received: July 31, 2014

Revised: July 31, 2015

Published: October 2015

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Abstract

Nonlinear Schrödinger equations with inverse-square potentials are considered in space dimension $N=2$. Stricharz estimates for (NLS)a are shown by Burq, Planchon, Stalker and Tahvildar-Zadeh [1] even when $N=2$. Here there seems not to be the study of solvability of (NLS)a when dimension is two. By virtue of the Hardy inequality the solvability is proved in Okazawa-Suzuki-Yokota, [3,4] if $N\ge 3$. Although strongly singular potential $a|x|^{-2}$ is available and the energy space is not exactly $H^{1}$ in (NLS)a, we can apply the energy methods established by Okazawa-Suzuki-Yokota [4].

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Mathematics Subject Classification: Primary: 35Q55, 35Q40; Secondary: 81Q15.

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References

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