Uniqueness conditions for B > 0

Identify general conditions on the potential V(x) ensuring that the Liouville equation −Δu(x) = 4π B V(x) e^{u(x)} in R^2 admits a unique solution when B > 0. Specify structural, regularity, or asymptotic assumptions on V that preclude multiple solutions beyond the radially symmetric setting.

Background

Theorem 1.4 proves uniqueness for radially symmetric solutions under monotonicity and decay assumptions on V, but uniqueness in the general (possibly non‑radial) case remains unresolved. The authors note that uniqueness can fail in certain symmetric situations (e.g., conformal invariance when V ≡ 1), and examples exist with non‑radial solutions even when some radial conditions are met.

This open question asks for a comprehensive set of criteria on V guaranteeing uniqueness for B > 0, extending beyond the radial framework of Theorem 1.4.