Schrödinger equation with multiparticle potential and critical nonlinearity

Jan Chabrowski, Andrzej Szulkin, Michael Willem

DOI: http://dx.doi.org/10.12775/TMNA.2009.038

Abstract


We study the existence and non-existence of ground states
for the Schrödinger equations
$-\Delta u -\lambda\sum_{i< j}u/|x_i-x_j|^2 = |u|^{2^*-2}u$,
$x=(x_1,\ldots,x_m)\in {\mathbb R}^{mN}$,
and
$-\Delta u -\lambda u/|y|^2 = |u|^{2^*-2}u,\quad x=(y,z)\in {\mathbb R}^N$.
In both cases we assume $\lambda\ne 0$ and $\lambda< \overline\lambda$, where
$\overline\lambda$ is the Hardy constant corresponding to the problem.

Keywords


Schrödinger equation; multiparticle potential; Hardy inequality; ground state; concentration-compactness

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