A Harnack type inequality for singular Liouville type equations

We obtain a Harnack type inequality for solutions of the Liouville type equation, {equation} -Δ u=|x|2αK(x)eᵘ \,\,\, Ω, {equation} where α∈(-1,0), Ω is a bounded domain in R² and K satisfies, {equation} 0<a≤ K(x)≤ b<+∞. {equation} This is a generalization to the singular case of a result by C.C. Chen and C.S. Lin [Comm. An. Geom. 1998], which considered the regular case α=0. Part of the argument of Chen-Lin can be adapted to the singular case by means of an isoperimetric inequality for surfaces with conical singularities. However, the case α∈(-1,0) turns out to be more delicate, due to the lack of traslation invariance of the singular problem, which requires a different approach.