Geodesic
Critical case of nonlinear Schrödinger equations with inverse-square potentials on bounded domains
Mathematica Bohemica, Tome 139 (2014) no. 2, pp. 231-238

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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 $H_{0}^{1}(\Omega )\subset D(P_{a(N)}^{1/2}) \subset H^{s}(\Omega )$ ($s1$). Therefore we can construct global weak solutions to (NLS)$_{a}$ on $\Omega $ by the energy methods.
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 $H_{0}^{1}(\Omega )\subset D(P_{a(N)}^{1/2}) \subset H^{s}(\Omega )$ ($s1$). Therefore we can construct global weak solutions to (NLS)$_{a}$ on $\Omega $ by the energy methods.
DOI : 10.21136/MB.2014.143851
Classification : 35A01, 35A23, 35D30, 35Q40, 35Q55, 81Q15
Keywords: energy method; nonlinear Schrödinger equation; inverse-square potential; Hardy-Poincaré inequality 
Suzuki, Toshiyuki. Critical case of nonlinear Schrödinger equations with inverse-square potentials on bounded domains. Mathematica Bohemica, Tome 139 (2014) no. 2, pp. 231-238. doi: 10.21136/MB.2014.143851
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