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

Abstract

We consider generic 2 × 2 singular Liouville systemswhere Ω ∋ 0 is a smooth bounded domain in R 2 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 (1) 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.