The non-local mean-field equation on an interval

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.