This note studies the inhomogeneous generalized Hartree equation \(\begin{aligned} i\dot{u}+\Delta u=\pm |x|^{-\rho }|u|^{p-2}(J_\gamma *|\cdot |^{-\rho }|u|^p)u,\quad \rho>0,\, p>2. \end{aligned}\) The goal of this work is two-fold. First, one obtains the existence of a local solution in \(C_T(H^{s_c})\) , where the critical Sobolev exponent is given by the equality \(\lambda ^\frac{2-2\rho +\gamma }{2(p-1)}\Vert u_0(\lambda \cdot )\Vert _{\dot{H}^{s_c}}=\Vert u_0\Vert _{\dot{H}^{s_c}}\) . Second, one investigates the uniqueness of critical solutions in \(C_T(H^{s_c})\) . Indeed, since one uses a fixed point argument in some Strichartz spaces, the uniqueness in the energy space is not trivial. In fact, the technique used in order to obtain the existence of a local sub-critical solution, which consists to divide the integrals on the unit centered ball of \(\mathbb {R}^N\) and it’s complementary is no more sufficient to conclude. To overcome this difficulty, one uses a fixed point argument with Strichartz estimates in some suitable Lorentz spaces, which enables us to handle the inhomogeneous term by the fact that \(|x|^{-\rho }\in L^{\frac{N}{\rho },\infty }\) .
