# American Institute of Mathematical Sciences

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

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].
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, J. Funct. Anal., 203 (2003), 519-549. [2] T.Ogawa, A proof of Trudinger's inequality and its application to nonlinear Schrödinger equations, Nonlinear Anal., 14 (1990), 765-769. [3] N.Okazawa, T.Suzuki, T.Yokota, Cauchy problem for nonlinear Schrödinger equations with inverse-square potentials, Appl. Anal., 91 (2012), 1605-1629. [4] N.Okazawa, T.Suzuki, T.Yokota, Energy methods for abstract nonlinear Schrödinger equations, Evol. Equ. Control Theory, 1 (2012), 337-354. [5] T.Suzuki, Energy methods for Hartree type equation with inverse-square potentials, Evol. Equ. Control Theory, 2 (2013), 531-542. [6] T.Suzuki, Critical case of nonlinear Schrödinger equations with inverse-square potentials on bounded domains, Math. Bohemica, 139 (2014), 231-238. [7] T.Suzuki, Solvability of nonlinear Schrödinger equations with some critical singular potential via generalized Hardy-Rellich inequalities, Funkcial. Ekvac., to appear. [8] L.Wei, Z.Feng, Isolated singularity for semilinear elliptic equations, Discrete and Continuous Dynamical System-A, 35 (2015), 3239-3252.

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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, J. Funct. Anal., 203 (2003), 519-549. [2] T.Ogawa, A proof of Trudinger's inequality and its application to nonlinear Schrödinger equations, Nonlinear Anal., 14 (1990), 765-769. [3] N.Okazawa, T.Suzuki, T.Yokota, Cauchy problem for nonlinear Schrödinger equations with inverse-square potentials, Appl. Anal., 91 (2012), 1605-1629. [4] N.Okazawa, T.Suzuki, T.Yokota, Energy methods for abstract nonlinear Schrödinger equations, Evol. Equ. Control Theory, 1 (2012), 337-354. [5] T.Suzuki, Energy methods for Hartree type equation with inverse-square potentials, Evol. Equ. Control Theory, 2 (2013), 531-542. [6] T.Suzuki, Critical case of nonlinear Schrödinger equations with inverse-square potentials on bounded domains, Math. Bohemica, 139 (2014), 231-238. [7] T.Suzuki, Solvability of nonlinear Schrödinger equations with some critical singular potential via generalized Hardy-Rellich inequalities, Funkcial. Ekvac., to appear. [8] L.Wei, Z.Feng, Isolated singularity for semilinear elliptic equations, Discrete and Continuous Dynamical System-A, 35 (2015), 3239-3252.
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