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: $$\{arrayll\mathrm{\Delta }{u}_i+\sum _{j=1}^n{a}_{ij}\mathrm{\mid }x{\mathrm{\mid }}^{{\beta }_j}{e}^{u_j(x)}=0,{\mathbb{R}}^2,i=1,\mathrm{.}\mathrm{.}\mathrm{.},n{\int }_{{\mathbb{R}}^2}\mathrm{\mid }x{\mathrm{\mid }}^{{\beta }_i}{e}^{u_i(x)}dx<\mathrm{\infty },i=1,\mathrm{.}\mathrm{.}\mathrm{.},narray\mathrm{.}$$ where ${\beta }_1,\mathrm{.}\mathrm{.}\mathrm{.},{\beta }_n$ 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.