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

TL;DR

This paper develops a unified framework for analyzing blow-up phenomena in two-dimensional singular Liouville systems, constructing solutions with prescribed blow-up behavior and local mass, using explicit formulas involving Chebyshev polynomials.

Contribution

It introduces a novel method to construct solutions with specific blow-up characteristics in singular Liouville systems, linking blow-up values to Chebyshev polynomials.

Findings

Constructed solutions with prescribed blow-up at the origin.

Identified conditions for finitely or infinitely many blow-up values.

Derived explicit formulas for blow-up values using Chebyshev polynomials.

Abstract

We consider generic 2 x 2 singular Liouville systems on a smooth bounded domain in the plane having some symmetry with respect to the origin. We construct a family of solutions to which blow-up at the origin and whose local mass at the origin is a given quantity depending on the parameters of the system. We can get either finitely many possible blow-up values of the local mass or infinitely many. The blow-up values are produced using an explicit formula which involves Chebyshev polynomials.