Symmetry and monotonicity of singular solutions for the Hartree equation

TL;DR

This paper investigates positive singular solutions of a nonlocal Hartree equation, establishing their symmetry and monotonicity around the singular set, and demonstrating the existence of such solutions with these properties.

Contribution

It proves symmetry and monotonicity of singular solutions for the Hartree equation and constructs an example of such solutions, extending understanding of their structure.

Findings

Solutions are symmetric and monotone with respect to the singular set.

Existence of singular solutions with symmetry and monotonicity properties.

Application of moving plane methods to nonlocal equations.

Abstract

In this paper we are concerned with positive singular solutions of the following nonlocal Hartree equation $$-\Delta u\!=\Big( \int_{\mathbb{R}^N\setminus \Gamma}\frac{F(u(y))}{|x-y|^\mu}dy \Big)f (u(x)), \quad x\in \mathbb{R}^N\setminus\Gamma,$$ where $F$ is the primitive of $f$ and $\Gamma$ is the singular set. Under suitable assumptions, we prove that $u$ is symmetric and monotone with respect to the singular set by using moving plane methods. Furthermore, we complement this study by showing the existence, for a model problem, of a singular solution with the desired properties.