### Schrödinger equation with multiparticle potential and critical nonlinearity

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.

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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