Orbital Stability for the Schr\"{o}dinger Operator Involving Inverse Square Potential

Abstract

In this paper we prove the existence of orbitally stable standing waves for the critical Schr\"{o}dinger operator, involving potential of the form $\left(\frac{N-2}{2}\right)^2|x|^{-2}$. The approach, being purely variational, is based on the precompactness of any minimizing sequence with respect to the associated energy. Moreover, we discuss the case of the presence of a Hardy energy term, in conjunction with the behavior of the standing waves at the singularity.