American Institute of Mathematical Sciences

Advanced
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, 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

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, 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

[1]

Toshiyuki Suzuki. Energy methods for Hartree type equations with inverse-square potentials. Evolution Equations & Control Theory, 2013, 2 (3) : 531-542. doi: 10.3934/eect.2013.2.531
[2]

Toshiyuki Suzuki. Scattering theory for semilinear Schrödinger equations with an inverse-square potential via energy methods. Evolution Equations & Control Theory, 2019, 8 (2) : 447-471. doi: 10.3934/eect.2019022
[3]

Veronica Felli, Ana Primo. Classification of local asymptotics for solutions to heat equations with inverse-square potentials. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 65-107. doi: 10.3934/dcds.2011.31.65
[4]

Noboru Okazawa, Toshiyuki Suzuki, Tomomi Yokota. Energy methods for abstract nonlinear Schrödinger equations. Evolution Equations & Control Theory, 2012, 1 (2) : 337-354. doi: 10.3934/eect.2012.1.337
[5]

Suna Ma, Huiyuan Li, Zhimin Zhang. Novel spectral methods for Schrödinger equations with an inverse square potential on the whole space. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1589-1615. doi: 10.3934/dcdsb.2018221
[6]

Gisèle Ruiz Goldstein, Jerome A. Goldstein, Abdelaziz Rhandi. Kolmogorov equations perturbed by an inverse-square potential. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 623-630. doi: 10.3934/dcdss.2011.4.623
[7]

Rowan Killip, Changxing Miao, Monica Visan, Junyong Zhang, Jiqiang Zheng. The energy-critical NLS with inverse-square potential. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3831-3866. doi: 10.3934/dcds.2017162
[8]

Tiziana Durante, Abdelaziz Rhandi. On the essential self-adjointness of Ornstein-Uhlenbeck operators perturbed by inverse-square potentials. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 649-655. doi: 10.3934/dcdss.2013.6.649
[9]

Xing Cheng, Ze Li, Lifeng Zhao. Scattering of solutions to the nonlinear Schrödinger equations with regular potentials. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 2999-3023. doi: 10.3934/dcds.2017129
[10]

Yongsheng Jiang, Huan-Song Zhou. A sharp decay estimate for nonlinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1723-1730. doi: 10.3934/cpaa.2010.9.1723
[11]

Rémi Carles. Global existence results for nonlinear Schrödinger equations with quadratic potentials. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 385-398. doi: 10.3934/dcds.2005.13.385
[12]

Zaihui Gan, Boling Guo, Jian Zhang. Blowup and global existence of the nonlinear Schrödinger equations with multiple potentials. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1303-1312. doi: 10.3934/cpaa.2009.8.1303
[13]

Zaihui Gan. Cross-constrained variational methods for the nonlinear Klein-Gordon equations with an inverse square potential. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1541-1554. doi: 10.3934/cpaa.2009.8.1541
[14]

Shalva Amiranashvili, Raimondas  Čiegis, Mindaugas Radziunas. Numerical methods for a class of generalized nonlinear Schrödinger equations. Kinetic & Related Models, 2015, 8 (2) : 215-234. doi: 10.3934/krm.2015.8.215
[15]

Hideo Takaoka. Energy transfer model for the derivative nonlinear Schrödinger equations on the torus. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5819-5841. doi: 10.3934/dcds.2017253
[16]

Zhongwei Tang. Least energy solutions for semilinear Schrödinger equations involving critical growth and indefinite potentials. Communications on Pure & Applied Analysis, 2014, 13 (1) : 237-248. doi: 10.3934/cpaa.2014.13.237
[17]

Hengguang Li, Jeffrey S. Ovall. A posteriori eigenvalue error estimation for a Schrödinger operator with inverse square potential. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1377-1391. doi: 10.3934/dcdsb.2015.20.1377
[18]

Fabrice Planchon, John G. Stalker, A. Shadi Tahvildar-Zadeh. $L^p$ Estimates for the wave equation with the inverse-square potential. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 427-442. doi: 10.3934/dcds.2003.9.427
[19]

Fabrice Planchon, John G. Stalker, A. Shadi Tahvildar-Zadeh. Dispersive estimate for the wave equation with the inverse-square potential. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1387-1400. doi: 10.3934/dcds.2003.9.1387
[20]

Weiwei Ao, Juncheng Wei, Wen Yang. Infinitely many positive solutions of fractional nonlinear Schrödinger equations with non-symmetric potentials. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5561-5601. doi: 10.3934/dcds.2017242

 Impact Factor: 

Tools

Metrics

  • PDF downloads (32)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences