A unified approach of blow-up phenomena for two-dimensional singular Liouville systems

We consider generic 2 × 2 singular Liouville systems [Equation presented here] where Ω∈ 0 is a smooth bounded domain in R2 possibly having some symmetry with respect to the origin, δ0 is the Dirac mass at 0, λ1, λ2 are small positive parameters and a, b, α1, α2 > 0. We construct a family of solutions to (∗) which blow up at the origin as λ1 → 0 and λ2 → 0 and whose local mass at the origin is a given quantity depending on a, b, α1, α2. In particular, if ab < 4 we get finitely many possible blow-up values of the local mass, whereas if ab ≥ 4 we get infinitely many. The blow-up values are produced using an explicit formula which involves Chebyshev polynomials.