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Title: Critical case of nonlinear Schrödinger equations with inverse-square potentials on bounded domains (English)
Author: Suzuki, Toshiyuki
Language: English
Journal: Mathematica Bohemica
ISSN: 0862-7959 (print)
ISSN: 2464-7136 (online)
Volume: 139
Issue: 2
Year: 2014
Pages: 231-238
Summary lang: English
Category: math
Summary: Nonlinear Schrödinger equations (NLS)$_{a}$ with strongly singular potential $a|x|^{-2}$ on a bounded domain $\Omega $ are considered. If $\Omega =\mathbb {R}^{N}$ and $a>-(N-2)^{2}/4$, then the global existence of weak solutions is confirmed by applying the energy methods established by N. Okazawa, T. Suzuki, T. Yokota (2012). Here $a=-(N-2)^{2}/4$ is excluded because $D(P_{a(N)}^{1/2})$ is not equal to $H^{1}(\mathbb R^{N})$, where $P_{a(N)}:=-\Delta -(N-2)^{2}/(4|x|^{2})$ is nonnegative and selfadjoint in $L^{2}(\mathbb R^{N})$. On the other hand, if $\Omega $ is a smooth and bounded domain with $0\in \Omega $, the Hardy-Poincaré inequality is proved in J. L. Vazquez, E. Zuazua (2000). Hence we can see that $H_{0}^{1}(\Omega )\subset D(P_{a(N)}^{1/2}) \subset H^{s}(\Omega )$ ($s<1$). Therefore we can construct global weak solutions to (NLS)$_{a}$ on $\Omega $ by the energy methods. (English)
Keyword: energy method
Keyword: nonlinear Schrödinger equation
Keyword: inverse-square potential
Keyword: Hardy-Poincaré inequality
MSC: 35A01
MSC: 35A23
MSC: 35D30
MSC: 35Q40
MSC: 35Q55
MSC: 81Q15
idZBL: Zbl 06362255
idMR: MR3238836
DOI: 10.21136/MB.2014.143851
Date available: 2014-07-14T08:17:51Z
Last updated: 2020-07-29
Stable URL:
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