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

TL;DR

This paper establishes the existence and orbital stability of standing waves for a fractional Hartree equation with unbounded potentials, using compact embedding and concentration compactness methods.

Contribution

It proves the existence of ground states in a fractional energy space and demonstrates their orbital stability, extending previous results to unbounded potentials.

Findings

Existence of ground states in fractional Sobolev space

Orbital stability of standing waves established

Compact embedding in the energy space proved

Abstract

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