Semiclassical analysis for pseudo-relativistic Hartree equations

TL;DR

This paper investigates the semiclassical limit of a pseudo-relativistic Hartree equation involving a nonlocal convolution term, providing insights into the behavior of solutions as the semiclassical parameter approaches zero.

Contribution

It introduces a detailed analysis of the semiclassical limit for the pseudo-relativistic Hartree equation with general parameters, extending previous studies to a broader class of nonlocal equations.

Findings

Characterization of solution behavior in the semiclassical limit

Existence of concentrating solutions around minima of the potential

Extension to Coulomb and other convolution kernels

Abstract

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.