This work investigates the Cauchy problem for a focusing inhomogeneous nonlinear Hartree equation in the energy space \(H^1_{rd}(\mathbb R^N)\) . The goal is three folds. First, one gives a sharp Gagliardo-Nirenberg type inequality adapted to the evolution problem. Second, one proves a local well-posedness result in the energy space. Third, one obtains a dichotomy of global existence and scattering versus blow-up of energy solutions in the inter-critical regime. The main novelty is to deal with the unbounded inhomogeneous term \(|x|^b\) for certain \(b>0\) and a non-local source term. For the scattering, one uses the method of Dodson and Murphy [Proc Amer Math Soc. 2017; 145 (11): 4859-4867]. This method is based on Morawetz estimates and a Tao’s scattering criterion [Dyn Partial Differ Equ. 2004; 1(1): 1-47]. The main ingredients are Strichartz estimates and some Strauss type inequalities. The threshold is expressed in term of non-conserved quantities in the spirit of Dinh [Discr Cont Dyn Syst. 2020; 40 (11): 6441-6471]. The radial assumption is used in order to estimate the potential energy taking account of the unbounded inhomogeneous term \(|x|^b\) for \(b>0\) .
