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

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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].
    Mathematics Subject Classification: Primary: 35Q55, 35Q40; Secondary: 81Q15.

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  • [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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