Schrödinger equation with multiparticle potential and critical nonlinearity

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 $-\Delta u -\lambda u/|y|^2 = |u|^{2^*-2}u$, $x=(y,z)\in {\mathbb{R}}^N$. In both cases we assume $\lambda\ne 0$ and $\lambda< \overline\lambda$, where $\overline\lambda$ is the Hardy constant corresponding to the problem.