Small-energy regularity for stationary weak solutions of the Liouville equation

We establish small-energy regularity for stationary weak solutions of the Liouville equation $-\mathrm{\Delta }u=e^u$ in every dimension. It solves a problem left open in Da Lio [Commun. Partial Differ. Equations, 2008] and posed explicitly again by Da Lio and Hyder in [Commun. Contemp. Math. 2025]. The proof relies on the analysis of concentration phenomena for almost stationary fields with structure constraints.