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

Abstract

We consider generic $2\times 2$ singular Liouville systems $$ (\star)\quad \left{ \begin{array}{ll} -\Delta u\_1=2\lambda\_1 ,e^{u\_1}-a\lambda\_2 ,e^{u\_2}-2\pi (\alpha\_1-2)\delta\_0& \text{in }\Omega, \ -\Delta u\_2=2\lambda\_2 ,e^{u\_2}-b\lambda\_1 ,e^{u\_1}-2\pi (\alpha\_2-2)\delta\_0&\text{in }\Omega,\ u\_1=u\_2=0&\text{on }\partial\Omega,\end{array}\right. $$ 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.