# Article

 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: [1] Burq, N., Planchon, F., Stalker, J. G., Tahvildar-Zadeh, A. S.: Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential.J. Funct. Anal. 203 (2003), 519-549. Zbl 1030.35024, MR 2003358, 10.1016/S0022-1236(03)00238-6 Reference: [2] Burq, N., Planchon, F., Stalker, J. G., Tahvildar-Zadeh, A. S.: Strichartz estimates for the wave and Schrödinger equations with potentials of critical decay.Indiana Univ. Math. J. 53 (2004), 1665-1680. Zbl 1084.35014, MR 2106340, 10.1512/iumj.2004.53.2541 Reference: [3] Cazenave, T.: An Introduction to Nonlinear Schrödinger Equation.Textos de Métodos Matemáticos 22 Instituto de Matemática, Universidade Federal do Rio de Janeiro, Rio de Janeiro (1989). Reference: [4] Cazenave, T.: Semilinear Schrödinger Equations.Courant Lecture Notes in Mathematics 10 American Mathematical Society, Providence, Courant Institute of Mathematical Sciences, New York (2003). Zbl 1055.35003, MR 2002047 Reference: [5] Ginibre, J., Velo, G.: On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case.J. Funct. Anal. 32 (1979), 1-32. Zbl 0396.35028, MR 0533218, 10.1016/0022-1236(79)90076-4 Reference: [6] Kato, T.: On nonlinear Schrödinger equations.Ann. Inst. Henri Poincaré, Phys. Théor. 46 (1987), 113-129. Zbl 0632.35038, MR 0877998 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: [11] Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators.North-Holland Mathematical Library 18 North-Holland, Amsterdam (1978). Zbl 0387.46033, MR 0503903 Reference: [12] Vazquez, J. L., Zuazua, E.: The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential.J. Funct. Anal. 173 (2000), 103-153. Zbl 0953.35053, MR 1760280, 10.1006/jfan.1999.3556 .

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