| 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:
|
http://hdl.handle.net/10338.dmlcz/143851 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
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| Reference:
|
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| Reference:
|
[7] Okazawa, N.: $L^{p}$-theory of Schrödinger operators with strongly singular potentials.Jap. J. Math., New Ser. 22 (1996), 199-239. MR 1432373, 10.4099/math1924.22.199 |
| Reference:
|
[8] Okazawa, N., Suzuki, T., Yokota, T.: Cauchy problem for nonlinear Schrödinger equations with inverse-square potentials.Appl. Anal. 91 (2012), 1605-1629. Zbl 1246.35189, MR 2959550, 10.1080/00036811.2011.631914 |
| Reference:
|
[9] Okazawa, N., Suzuki, T., Yokota, T.: Energy methods for abstract nonlinear Schrödinger equations.Evol. Equ. Control Theory 1 (2012), 337-354. Zbl 1283.35128, MR 3085232, 10.3934/eect.2012.1.337 |
| Reference:
|
[10] Suzuki, T.: Energy methods for Hartree type equations with inverse-square potentials.Evol. Equ. Control Theory 2 (2013), 531-542. Zbl 1282.35358, MR 3093229, 10.3934/eect.2013.2.531 |
| Reference:
|
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| Reference:
|
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| . |
