Energy methods for Hartree type equations with inverse-square potentials

Abstract

Nonlinear Schrödinger equationswith nonlocal nonlinearities described by integral operatorsare considered.This generalizes usual Hartree type equations (HE)$_{0}$.We construct weak solutions to (HE)$_{a}$, $a\neq 0$,even if the kernel is of non-convolution type.The advantage of our methods is the applicabilityto the problem with strongly singular potential $a|x|^{-2}$as a term in the linear part andwith critical nonlinearity.