# The non-local mean-field equation on an interval

## TL;DR

This paper studies a fractional mean-field equation on an interval, establishing existence conditions related to the parameter  and developing tools for analyzing non-local equations with potential extensions.

## Contribution

It proves existence of solutions for the fractional mean-field equation on an interval precisely when <2, and introduces analytical tools applicable to higher-order non-local mean field equations.

## Key findings

- Existence of solutions if and only if <2.
- Analysis of blow-up sequences for solutions.
- Development of tools extendable to higher-order non-local equations.

## Abstract

We consider the fractional mean-field equation on the interval $I=(-1,1)$ $$(-\Delta)^\frac{1}{2} u=\rho\frac{e^{u}}{\int_{I}e^{u}dx},$$ subject to Dirichlet boundary conditions, and prove that existence holds if and only if $\rho <2\pi$. This requires the study of blowing-up sequences of solutions. We provide a series of tools in particular which can be used (and extended) to higher-order mean field equations of non-local type.