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

Abstract

We consider generic $2\times 2$ singular Liouville systems

where $\Omega \ni 0$ is a smooth bounded domain in ${\mathbb{R}}^2$ possibly having some symmetry with respect to the origin, ${\delta }_0$ is the Dirac mass at $0,$ ${\lambda }_1,{\lambda }_2$ are small positive parameters and $a,b,{\alpha }_1,{\alpha }_2>0$.

We construct a family of solutions to $(\star )$ which blow up at the origin as ${\lambda }_1\to 0$ and ${\lambda }_2\to 0$ and whose local mass at the origin is a given quantity depending on $a,b,{\alpha }_1,{\alpha }_2$.

In particular, if $ab<4$ we get finitely many possible blow-up values of the local mass, whereas if $ab\ge 4$ we get infinitely many. The blow-up values are produced using an explicit formula which involves Chebyshev polynomials.