Critical case of nonlinear Schrödinger equations with inverse-square potentials on bounded domains

Suzuki, Toshiyuki. "Critical case of nonlinear Schrödinger equations with inverse-square potentials on bounded domains." Mathematica Bohemica 139.2 (2014): 231-238. <http://eudml.org/doc/261912>.

@article{Suzuki2014,
abstract = {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.},
author = {Suzuki, Toshiyuki},
journal = {Mathematica Bohemica},
keywords = {energy method; nonlinear Schrödinger equation; inverse-square potential; Hardy-Poincaré inequality; energy method; nonlinear Schrödinger equation; inverse-square potential; Hardy-Poincaré inequality},
language = {eng},
number = {2},
pages = {231-238},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Critical case of nonlinear Schrödinger equations with inverse-square potentials on bounded domains},
url = {http://eudml.org/doc/261912},
volume = {139},
year = {2014},
}

TY - JOUR
AU - Suzuki, Toshiyuki
TI - Critical case of nonlinear Schrödinger equations with inverse-square potentials on bounded domains
JO - Mathematica Bohemica
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 139
IS - 2
SP - 231
EP - 238
AB - 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.
LA - eng
KW - energy method; nonlinear Schrödinger equation; inverse-square potential; Hardy-Poincaré inequality; energy method; nonlinear Schrödinger equation; inverse-square potential; Hardy-Poincaré inequality
UR - http://eudml.org/doc/261912
ER -