Semiclassical analysis for pseudo-relativistic Hartree equations

Abstract

In this paper we study the semiclassical limit for the pseudo-relativistic Hartree equation $\sqrt{-{\epsilon }^2\Delta +m^2}u+Vu=(I_{\alpha }\ast |u{|}^p)|u{|}^{p-2}u$ in ${\mathbb{R}}^N$ where $m>0$, $2\le p<\frac{2N}{N-1}$, $V:{\mathbb{R}}^N\to \mathbb{R}$ 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.

Accepted for publication by Journal of Differential Equations

References in corpus (1)

• Classical limit for semi-relativistic Hartree systems