Classification of Radial Solutions to Liouville Systems with Singularities

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.