Revista Matemática Iberoamericana


Full-Text PDF (502 KB) | Metadata | Table of Contents | RMI summary
Volume 34, Issue 4, 2018, pp. 1867–1910
DOI: 10.4171/rmi/1047

Published online: 2018-12-17

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

Luca Battaglia[1] and Angela Pistoia[2]

(1) Università degli Studi Roma Tre, Italy
(2) Università di Roma "La Sapienza", Roma, Italy

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.

Keywords: Liouville systems, blow-up phenomena, tower of bubbles

Battaglia Luca, Pistoia Angela: A unified approach of blow-up phenomena for two-dimensional singular Liouville systems. Rev. Mat. Iberoam. 34 (2018), 1867-1910. doi: 10.4171/rmi/1047