Radial symmetry of all solutions for n > 0 under asymptotic behavior

Establish whether all solutions to the normalized Liouville equation −Δu(x) = 4π B |x|^{n} V(x) e^{u(x)} in R^2 with ∫_{R^2} |x|^{n} V(x) e^{u(x)} dx = 1 are radially symmetric when n > 0 and u satisfies the asymptotic condition lim_{|x|→∞} u(x)/log|x| = −B/2.

Background

Theorem 1.4 proves uniqueness of the radially symmetric solution under assumptions including monotonicity and decay of V. For n = 0, known results imply that all solutions with the specified asymptotics are radial, but for n > 0 the authors cannot verify this.

This problem seeks a symmetry classification of all solutions under the prescribed asymptotic behavior, extending classical symmetry results to weighted settings with |x|^{n}.