Semiclassical analysis for pseudo-relativistic Hartree equations

Abstract: In this paper we study the semiclassical limit for the pseudo-relativistic Hartree equation $$\sqrt{-\varepsilon² \Delta + m^2}u + V u = (I_\alpha * |u|^{p}) |u|<sup>{p-2}u$$ in $\mathbb{R}<sup>N$ where $m&gt;0$, $2 \leq p &lt; \frac{2N}{N-1}$, $V \colon \mathbb{R}^N \to \mathbb{R}$ is an external scalar potential, $$I_\alpha (x) = \frac{c_{N,\alpha}}{|x|<sup>{N-\alpha}}$$ is a convolution kernel, $c_{N,\alpha}$ is a positive constant and $(N-1)p-N&lt;\alpha &lt;N$. For $N=3$, $\alpha=p=2$, our equation becomes the pseudo-relativistic Hartree equation with Coulomb kernel.