We prove the existence of the set of ground states in a suitable energy space $\Sigma^s=\{ u: \int_{ \mathbb{ R}^N} \bar{ u}(-\Delta+m^2)^s u+V |u|^2<\infty\}$, $s\in (0,\frac{ N}{ 2})$ for the mass-subcritical nonlinear fractional Hartree equation with unbounded potentials. As a consequence we obtain, as a priori result, the orbital stability of the set of standing waves. The main ingredient is the observation that $\Sigma^s$ is compactly embedded in $L^2$. This enables us to apply the concentration compactness argument in the works of Cazenave-Lions and Zhang, namely, relative compactness for any minimizing sequence in the energy space.
Categories: math.AP math-ph math.MP