American Institute of Mathematical Sciences

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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

Toshiyuki Suzuki 1,

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].
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, J. Funct. Anal., 203 (2003), 519-549. Google Scholar

[2]

T.Ogawa, A proof of Trudinger's inequality and its application to nonlinear Schrödinger equations, Nonlinear Anal., 14 (1990), 765-769. Google Scholar

[3]

N.Okazawa, T.Suzuki, T.Yokota, Cauchy problem for nonlinear Schrödinger equations with inverse-square potentials, Appl. Anal., 91 (2012), 1605-1629. Google Scholar

[4]

N.Okazawa, T.Suzuki, T.Yokota, Energy methods for abstract nonlinear Schrödinger equations, Evol. Equ. Control Theory, 1 (2012), 337-354. Google Scholar

[5]

T.Suzuki, Energy methods for Hartree type equation with inverse-square potentials, Evol. Equ. Control Theory, 2 (2013), 531-542. Google Scholar

[6]

T.Suzuki, Critical case of nonlinear Schrödinger equations with inverse-square potentials on bounded domains, Math. Bohemica, 139 (2014), 231-238. 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), 3239-3252. Google Scholar

show all references

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. Google Scholar

[2]

T.Ogawa, A proof of Trudinger's inequality and its application to nonlinear Schrödinger equations, Nonlinear Anal., 14 (1990), 765-769. Google Scholar

[3]

N.Okazawa, T.Suzuki, T.Yokota, Cauchy problem for nonlinear Schrödinger equations with inverse-square potentials, Appl. Anal., 91 (2012), 1605-1629. Google Scholar

[4]

N.Okazawa, T.Suzuki, T.Yokota, Energy methods for abstract nonlinear Schrödinger equations, Evol. Equ. Control Theory, 1 (2012), 337-354. Google Scholar

[5]

T.Suzuki, Energy methods for Hartree type equation with inverse-square potentials, Evol. Equ. Control Theory, 2 (2013), 531-542. Google Scholar

[6]

T.Suzuki, Critical case of nonlinear Schrödinger equations with inverse-square potentials on bounded domains, Math. Bohemica, 139 (2014), 231-238. 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), 3239-3252. Google Scholar

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