Semiclassical analysis for pseudo-relativistic Hartree equations

In this paper we study the semiclassical limit for the pseudo-relativistic Hartree equation $\sqrt{-\varepsilon^2 \Delta + m^2}u + V u = (I_\alpha * |u|^{p}) |u|^{p-2}u$ in $\mathbb{R}^N$ where $m>0$, $2 \leq p < \frac{2N}{N-1}$, $V \colon \mathbb{R}^N \to \mathbb{R}$ is an external scalar potential, $I_\alpha (x) = \frac{c_{N,\alpha}}{|x|^{N-\alpha}}$ is a convolution kernel, $c_{N,\alpha}$ is a positive constant and $(N-1)p-N<\alpha <N$. For $N=3$, $\alpha=p=2$, our equation becomes the pseudo-relativistic Hartree equation with Coulomb kernel.