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energy method; nonlinear Schrödinger equation; inverse-square potential; Hardy-Poincaré inequality
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.
[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. DOI 10.1016/S0022-1236(03)00238-6 | MR 2003358 | Zbl 1030.35024
[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. DOI 10.1512/iumj.2004.53.2541 | MR 2106340 | Zbl 1084.35014
[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).
[4] Cazenave, T.: Semilinear Schrödinger Equations. Courant Lecture Notes in Mathematics 10 American Mathematical Society, Providence, Courant Institute of Mathematical Sciences, New York (2003). MR 2002047 | Zbl 1055.35003
[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. DOI 10.1016/0022-1236(79)90076-4 | MR 0533218 | Zbl 0396.35028
[6] Kato, T.: On nonlinear Schrödinger equations. Ann. Inst. Henri Poincaré, Phys. Théor. 46 (1987), 113-129. MR 0877998 | Zbl 0632.35038
[7] Okazawa, N.: $L^{p}$-theory of Schrödinger operators with strongly singular potentials. Jap. J. Math., New Ser. 22 (1996), 199-239. MR 1432373
[8] Okazawa, N., Suzuki, T., Yokota, T.: Cauchy problem for nonlinear Schrödinger equations with inverse-square potentials. Appl. Anal. 91 (2012), 1605-1629. DOI 10.1080/00036811.2011.631914 | MR 2959550 | Zbl 1246.35189
[9] Okazawa, N., Suzuki, T., Yokota, T.: Energy methods for abstract nonlinear Schrödinger equations. Evol. Equ. Control Theory 1 (2012), 337-354. DOI 10.3934/eect.2012.1.337 | MR 3085232 | Zbl 1283.35128
[10] Suzuki, T.: Energy methods for Hartree type equations with inverse-square potentials. Evol. Equ. Control Theory 2 (2013), 531-542. DOI 10.3934/eect.2013.2.531 | MR 3093229 | Zbl 1282.35358
[11] Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland Mathematical Library 18 North-Holland, Amsterdam (1978). MR 0503903 | Zbl 0387.46033
[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. DOI 10.1006/jfan.1999.3556 | MR 1760280 | Zbl 0953.35053
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