Classification of Radial Solutions to Liouville Systems with Singularities

TL;DR

This paper establishes the existence and uniqueness of radial solutions to a class of Liouville systems with singularities, and analyzes their properties depending on the signs of the singularity parameters.

Contribution

It proves existence, uniqueness, and radial symmetry of solutions for Liouville systems with singularities, extending previous results to broader parameter ranges.

Findings

Solutions are unique and radial when all eta_i are negative.

The linearized system around solutions is non-degenerate.

Existence and integrability conditions are established for solutions.

Abstract

Let $A=(a_{ij})_{n\times n}$ be a nonnegative, symmetric, irreducible and invertible matrix. We prove the existence and uniqueness of radial solutions to the following Liouville system with singularity: $$\{{array}{ll} \Delta u_i+\sum_{j=1}^n a_{ij}|x|^{\beta_j}e^{u_j(x)}=0,\quad \mathbb R^2, \quad i=1,...,n \int_{\mathbb R^2}|x|^{\beta_i}e^{u_i(x)}dx<\infty, \quad i=1,...,n {array}. $$ where $\beta_1,...,\beta_n$ are constants greater than -2. If all $\beta_i$s are negative we prove that all solutions are radial and the linearized system is non-degenerate.