Orbital Stability of Standing Waves for Fractional Hartree Equation with Unbounded Potentials

We prove the existence of the set of ground states in a suitable energy space ${\mathrm{\Sigma }}^s=\{u:{\int }_{{\mathbb{R}}^N}\bar{u}(-\mathrm{\Delta }+m^2{)}^{s}u+V\mathrm{\mid }u{\mathrm{\mid }}^2<\mathrm{\infty }\}$, $s\in (0,\frac{N}{2})$ 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 ${\mathrm{\Sigma }}^s$ 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.