MATHEMATICA BOHEMICA, Vol. 139, No. 2, pp. 231-238, 2014

Critical case of nonlinear Schrödinger equations with inverse-square potentials on bounded domains

Toshiyuki Suzuki

Toshiyuki Suzuki, Department of Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan, e-mail: t21.suzuki@gmail.com

Abstract: Nonlinear Schrödinger equations (NLS)$_a$ with strongly singular potential $a|x|^{-2}$ on a bounded domain $\Omega$ are considered. If $\Omega=\mathbb{R}^N$ and $a>-(N-2)^2/4$, then the global existence of weak solutions is confirmed by applying the energy methods established by N. Okazawa, T. Suzuki, T. Yokota (2012). Here $a=-(N-2)^2/4$ is excluded because $D(P_{a(N)}^{1/2})$ is not equal to $H^1(\mathbb R^N)$, where $P_{a(N)}:=-\Delta-(N-2)^2/(4|x|^2)$ is nonnegative and selfadjoint in $L^2(\mathbb R^N)$. On the other hand, if $\Omega$ is a smooth and bounded domain with $0\in\Omega$, the Hardy-PoincarĂ© inequality is proved in J. L. Vazquez, E. Zuazua (2000). Hence we can see that $H_0^1(\Omega)\subset D(P_{a(N)}^{1/2}) \subset H^s(\Omega)$ ($s<1$). Therefore we can construct global weak solutions to (NLS)$_a$ on $\Omega$ by the energy methods.

Keywords: energy method; nonlinear Schrödinger equation; inverse-square potential; Hardy-PoincarĂ© inequality

Classification (MSC 2010): 35Q55, 35Q40, 81Q15


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