Pure-mass regularity without a Dirichlet-energy bound

Determine whether, for every fixed dimension n≥3, there exist constants δ_n>0 and C_n<∞ such that every stationary weak solution u∈H^1(B_2) of −Δu=e^u with e^u∈L^1(B_2) and ∫_{B_2}e^u≤δ_n is smooth on B_1 and satisfies sup_{B_1}e^u≤C_n∫_{B_2}e^u, without any a priori bound on ∫_{B_2}|∇u|^2.

Background

The paper proves regularity from small total energy and, separately, proves that sufficiently small mass implies regularity when the Dirichlet energy is bounded by a prescribed constant. The admissible mass threshold in that result depends on the energy bound. The unresolved issue is whether the Dirichlet-energy hypothesis can be removed entirely, leaving only a dimension-dependent smallness assumption on the mass ∫_{B_2}eu.

The authors note that their examples demonstrate that small mass does not control the Dirichlet energy, including examples in which the energy becomes arbitrarily large, but these examples remain smooth and therefore do not settle the proposed regularity question. They state that they believe the answer is negative but have not found a counterexample.

References

For every fixed $n\geq3$, do there exist $\delta_n>0$ and $C_n<\infty$ such that every stationary weak solution $u\in H1(B_2)$ with $eu\in L1(B_2)$ satisfying

\int_{B_2}eu d x\leq\delta_n

is smooth on $B_1$ and satisfies $\sup_{B_1}eu\leq C_n\int_{B_2}eu$, without an a priori bound for $\int_{B_2}{\nabla u}2$?

We believe the answer is negative, although at present we still cannot find a counterexample.

— Small-energy regularity for stationary weak solutions of the Liouville equation  (2609.26026 - Fu et al., 22 Sep 2026) in Section 7, subsection “The role of the Dirichlet energy bound,” Question 7.1 (q:ba-puremass)