Nonlinear Schrödinger equations with inverse-square potentials in two dimensional space

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2015, 2015(special): 1019-1024. doi: 10.3934/proc.2015.1019

Nonlinear Schrödinger equations with inverse-square potentials in two dimensional space

1.	Department of Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601

Received August 2014 Revised August 2015 Published November 2015

Abstract
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Nonlinear Schrödinger equations with inverse-square potentials are considered in space dimension $N=2$. Stricharz estimates for (NLS)a are shown by Burq, Planchon, Stalker and Tahvildar-Zadeh [1] even when $N=2$. Here there seems not to be the study of solvability of (NLS)a when dimension is two. By virtue of the Hardy inequality the solvability is proved in Okazawa-Suzuki-Yokota, [3,4] if $N\ge 3$. Although strongly singular potential $a|x|^{-2}$ is available and the energy space is not exactly $H^{1}$ in (NLS)a, we can apply the energy methods established by Okazawa-Suzuki-Yokota [4].

Keywords: nonlinear Schrödinger equations, inverse-square potentials., Energy methods.

Mathematics Subject Classification: Primary: 35Q55, 35Q40; Secondary: 81Q1.

Citation: Toshiyuki Suzuki. Nonlinear Schrödinger equations with inverse-square potentials in two dimensional space. Conference Publications, 2015, 2015 (special) : 1019-1024. doi: 10.3934/proc.2015.1019

References:

[1]	N.Burq, F.Planchon, J.Stalker, A.S.Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger, equations with the inverse-square potential, 203 (2003), 519. Google Scholar
[2]	T.Ogawa, A proof of Trudinger's inequality and its application to nonlinear Schrödinger equations,, Nonlinear Anal., 14 (1990), 765. Google Scholar
[3]	N.Okazawa, T.Suzuki, T.Yokota, Cauchy problem for nonlinear Schrödinger equations with inverse-square potentials,, Appl. Anal., 91 (2012), 1605. Google Scholar
[4]	N.Okazawa, T.Suzuki, T.Yokota, Energy methods for abstract nonlinear Schrödinger equations,, Evol. Equ. Control Theory, 1 (2012), 337. Google Scholar
[5]	T.Suzuki, Energy methods for Hartree type equation with inverse-square potentials,, Evol. Equ. Control Theory, 2 (2013), 531. Google Scholar
[6]	T.Suzuki, Critical case of nonlinear Schrödinger equations with inverse-square potentials on bounded domains,, Math. Bohemica, 139 (2014), 231. Google Scholar
[7]	T.Suzuki, Solvability of nonlinear Schrödinger equations with some critical singular potential via generalized Hardy-Rellich inequalities,, Funkcial. Ekvac., (). Google Scholar
[8]	L.Wei, Z.Feng, Isolated singularity for semilinear elliptic equations,, Discrete and Continuous Dynamical System-A, 35 (2015), 3239. Google Scholar

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References:

[1]	N.Burq, F.Planchon, J.Stalker, A.S.Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger, equations with the inverse-square potential, 203 (2003), 519. Google Scholar
[2]	T.Ogawa, A proof of Trudinger's inequality and its application to nonlinear Schrödinger equations,, Nonlinear Anal., 14 (1990), 765. Google Scholar
[3]	N.Okazawa, T.Suzuki, T.Yokota, Cauchy problem for nonlinear Schrödinger equations with inverse-square potentials,, Appl. Anal., 91 (2012), 1605. Google Scholar
[4]	N.Okazawa, T.Suzuki, T.Yokota, Energy methods for abstract nonlinear Schrödinger equations,, Evol. Equ. Control Theory, 1 (2012), 337. Google Scholar
[5]	T.Suzuki, Energy methods for Hartree type equation with inverse-square potentials,, Evol. Equ. Control Theory, 2 (2013), 531. Google Scholar
[6]	T.Suzuki, Critical case of nonlinear Schrödinger equations with inverse-square potentials on bounded domains,, Math. Bohemica, 139 (2014), 231. Google Scholar
[7]	T.Suzuki, Solvability of nonlinear Schrödinger equations with some critical singular potential via generalized Hardy-Rellich inequalities,, Funkcial. Ekvac., (). Google Scholar
[8]	L.Wei, Z.Feng, Isolated singularity for semilinear elliptic equations,, Discrete and Continuous Dynamical System-A, 35 (2015), 3239. Google Scholar

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