MR Toshiyuki SUZUKI Toshiyuki SUZUKI Solvability of Nonlinear Schrödinger Equations with Some Critical Singular Potential via Generalized Hardy-Rellich Inequalities Funkcialaj Ekvacioj. Serio Internacia 59 2016 1--34 http://fe.math.kobe-u.ac.jp/FE/FullPapers/59-1/59_1.pdf http://www.ams.org/mathscinet-getitem?mr=MR Nonlinear Schrödinger equations with inverse-square potentials ${\bf (NLS)}_a$ are considered. Since the potential $|x|^{-2}$ is quite singular, the scaling argument does not work well. In view of the selfadjointness of $P_{a}:=-\Delta+a|x|^{-2}$, $a=a(N):=-(N-2)^{2}/4$ seems to be the threshold of the unique solvability. In fact, if $a>a(N)$, then the unique solvability for ${\bf (NLS)}_a$ is proved by the energy methods established by Okazawa-Suzuki-Yokota [12]. On the other hand, if $a<a(N)$, then $P_{a}$ is not nonnegative in $L^2(\mathbb{R}^N)$ and has a lot of selfadjoint extensions. Here $P_{a(N)}$ is nonnegative and selfadjoint in $L^2(\mathbb{R}^N)$ in the sense of form-sum. But the energy space $D((1+P_{a(N)})^{1/2})$ does not coincide with $H^1(\mathbb{R}^N)$. Thus we identify the energy space by applying generalized Hardy-Rellich inequalities. By virtue of the identification we can apply the energy methods and conclude the global solvability for ${\bf (NLS)}_a$ with $a=a(N)$, the critical coefficient. Moreover, the uniqueness can be shown by using the Strichartz estimates for $e^{-itP_{a(N)}}$ which is also proved. Nonlinear Schrödinger equation, Inverse-square potential, Hardy-Rellich inequality, Spherical harmonics decomposition, Fractional Sobolev spaces, Energy methods, Strichartz estimates. Primary 35Q55, 35Q40; Secondary 81Q15. 59-1 2016 Solvability of Nonlinear Schrödinger Equations with Some Critical Singular Potential via Generalized Hardy-Rellich Inequalities Toshiyuki SUZUKI Toshiyuki SUZUKI
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